From the output of pactl list sink-inputs, I need to grab the sink input number for VLC. Before that, I'm trying to extract the piece that contains the output for only VLC. All the methods that I thought would work have shortcomings. This is a sample output:
$ pactl list sink-inputs
Sink Input #1373
    Driver: protocol-native.c
    Owner Module: 9
    Client: 10350
    Sink: 0
    Sample Specification: float32le 2ch 44100Hz
    Channel Map: front-left,front-right
    Format: pcm, format.sample_format = "\"float32le\""  format.rate = "44100"  format.channels = "2"  format.channel_map = "\"front-left,front-right\""
    Corked: no
    Mute: no
    Volume: 0: 100% 1: 100%
            0: 0,00 dB 1: 0,00 dB
            balance 0,00
    Buffer Latency: 453287 usec
    Sink Latency: 19697 usec
    Resample method: copy
    Properties:
        media.role = "video"
        media.name = "audio stream"
        application.name = "VLC media player (LibVLC 2.1.5)"
        native-protocol.peer = "UNIX socket client"
        native-protocol.version = "28"
        application.id = "org.VideoLAN.VLC"
        application.version = "2.1.5"
        application.icon_name = "vlc"
        application.language = "pt_BR.UTF-8"
        application.process.id = "19965"
        application.process.machine_id = "948146522454ae6aa2bb8ed153f4bce4"
        application.process.session_id = "948146522454ae6aa2bb8ed153f4bce4-1431635199.85146-1790309877"
        application.process.user = "teresaejunior"
        application.process.host = "localhost"
        application.process.binary = "vlc"
        window.x11.display = ":0.0"
        module-stream-restore.id = "sink-input-by-media-role:video"
Sink Input #1378
    Driver: protocol-native.c
    Owner Module: 9
    Client: 10378
    Sink: 0
    Sample Specification: s16le 2ch 44100Hz
    Channel Map: front-left,front-right
    Format: pcm, format.sample_format = "\"s16le\""  format.rate = "44100"  format.channels = "2"  format.channel_map = "\"front-left,front-right\""
    Corked: no
    Mute: no
    Volume: 0:  87% 1:  87%
            0: -3,63 dB 1: -3,63 dB
            balance 0,00
    Buffer Latency: 989841 usec
    Sink Latency: 19572 usec
    Resample method: n/a
    Properties:
        media.name = "audio stream"
        application.name = "mplayer2"
        native-protocol.peer = "UNIX socket client"
        native-protocol.version = "28"
        application.process.id = "20093"
        application.process.user = "teresaejunior"
        application.process.host = "localhost"
        application.process.binary = "mplayer2"
        application.language = "C"
        window.x11.display = ":0.0"
        application.process.machine_id = "948146522454ae6aa2bb8ed153f4bce4"
        module-stream-restore.id = "sink-input-by-application-name:mplayer2"
Both awk '/^Sink/,/VLC/' and sed -n '/^Sink/,/VLC/p' grab the VLC part, but then grab the mplayer2 part too and go until the end of the output:
$ pactl list sink-inputs | awk '/^Sink/,/VLC/'
Sink Input #1373
    Driver: protocol-native.c
    Owner Module: 9
    Client: 10350
    Sink: 0
    Sample Specification: float32le 2ch 44100Hz
    Channel Map: front-left,front-right
    Format: pcm, format.sample_format = "\"float32le\""  format.rate = "44100"  format.channels = "2"  format.channel_map = "\"front-left,front-right\""
    Corked: no
    Mute: no
    Volume: 0: 100% 1: 100%
            0: 0,00 dB 1: 0,00 dB
            balance 0,00
    Buffer Latency: 437414 usec
    Sink Latency: 19666 usec
    Resample method: copy
    Properties:
        media.role = "video"
        media.name = "audio stream"
        application.name = "VLC media player (LibVLC 2.1.5)"
Sink Input #1379
    Driver: protocol-native.c
    Owner Module: 9
    Client: 10381
    Sink: 0
    Sample Specification: s16le 2ch 44100Hz
    Channel Map: front-left,front-right
    Format: pcm, format.sample_format = "\"s16le\""  format.rate = "44100"  format.channels = "2"  format.channel_map = "\"front-left,front-right\""
    Corked: no
    Mute: no
    Volume: 0:  87% 1:  87%
            0: -3,63 dB 1: -3,63 dB
            balance 0,00
    Buffer Latency: 980045 usec
    Sink Latency: 19563 usec
    Resample method: n/a
    Properties:
        media.name = "audio stream"
        application.name = "mplayer2"
        native-protocol.peer = "UNIX socket client"
        native-protocol.version = "28"
        application.process.id = "20093"
        application.process.user = "teresaejunior"
        application.process.host = "localhost"
        application.process.binary = "mplayer2"
        application.language = "C"
        window.x11.display = ":0.0"
        application.process.machine_id = "948146522454ae6aa2bb8ed153f4bce4"
        module-stream-restore.id = "sink-input-by-application-name:mplayer2"
grep -Poz '^Sink(?s).*?VLC' works, but if the VLC output should come after mplayer2, it would fail (a test with mplayer2 instead of VLC):
$ pactl list sink-inputs | grep -Poz '^Sink(?s).*?mplayer'
Sink Input #1373
    Driver: protocol-native.c
    Owner Module: 9
    Client: 10350
    Sink: 0
    Sample Specification: float32le 2ch 44100Hz
    Channel Map: front-left,front-right
    Format: pcm, format.sample_format = "\"float32le\""  format.rate = "44100"  format.channels = "2"  format.channel_map = "\"front-left,front-right\""
    Corked: no
    Mute: no
    Volume: 0: 100% 1: 100%
            0: 0,00 dB 1: 0,00 dB
            balance 0,00
    Buffer Latency: 441088 usec
    Sink Latency: 18159 usec
    Resample method: copy
    Properties:
        media.role = "video"
        media.name = "audio stream"
        application.name = "VLC media player (LibVLC 2.1.5)"
        native-protocol.peer = "UNIX socket client"
        native-protocol.version = "28"
        application.id = "org.VideoLAN.VLC"
        application.version = "2.1.5"
        application.icon_name = "vlc"
        application.language = "pt_BR.UTF-8"
        application.process.id = "19965"
        application.process.machine_id = "948146522454ae6aa2bb8ed153f4bce4"
        application.process.session_id = "948146522454ae6aa2bb8ed153f4bce4-1431635199.85146-1790309877"
        application.process.user = "teresaejunior"
        application.process.host = "localhost"
        application.process.binary = "vlc"
        window.x11.display = ":0.0"
        module-stream-restore.id = "sink-input-by-media-role:video"
Sink Input #1380
    Driver: protocol-native.c
    Owner Module: 9
    Client: 10396
    Sink: 0
    Sample Specification: s16le 2ch 44100Hz
    Channel Map: front-left,front-right
    Format: pcm, format.sample_format = "\"s16le\""  format.rate = "44100"  format.channels = "2"  format.channel_map = "\"front-left,front-right\""
    Corked: no
    Mute: no
    Volume: 0:  87% 1:  87%
            0: -3,63 dB 1: -3,63 dB
            balance 0,00
    Buffer Latency: 989841 usec
    Sink Latency: 18084 usec
    Resample method: n/a
    Properties:
        media.name = "audio stream"
        application.name = "mplayer
The desired output:
Sink Input #1373
    Driver: protocol-native.c
    Owner Module: 9
    Client: 10350
    Sink: 0
    Sample Specification: float32le 2ch 44100Hz
    Channel Map: front-left,front-right
    Format: pcm, format.sample_format = "\"float32le\""  format.rate = "44100"  format.channels = "2"  format.channel_map = "\"front-left,front-right\""
    Corked: no
    Mute: no
    Volume: 0: 100% 1: 100%
            0: 0,00 dB 1: 0,00 dB
            balance 0,00
    Buffer Latency: 441088 usec
    Sink Latency: 18159 usec
    Resample method: copy
    Properties:
        media.role = "video"
        media.name = "audio stream"
        application.name = "VLC media player (LibVLC 2.1.5)"