Assuming GNU grep, I would pick out the quoted strings and then join them together
grep -oP '".*?"' file | tr -d '\n' | sed 's/" *"//g'; echo
Result from your example (shortened for brevity)
"v=DKIM1; h=sha256; k=rsa; t=y; p=MIGfMA0GCSqG…QAB"
 Alternatively you could use perl,
perl -e '
    $_ = join(" ", (map {chomp; $_} <>));    # Read and join lines
    s/^.*?(".*").*$/$1/;                      # Strip leading and trailing text outside quoted strings
    s/"\s*"/ /g;                              # Merge adjacent quoted strings
    s/\s\s+/ /g;                              # Reduce multiple spaces to one
    print                                     # Output the result
' file
 
                