The Spok is dead...

Most of Gower Street isn't included in the TLRN, but as the A501 is part of it, the TLRN here extends into the side streets, so yes, it's TfL who are the highway authority responsible.

Report it here:
http://www.tfl.gov.uk/tfl/roadusers/reportastreetfault/newfault.aspx

Claim as above.

I've checked with one of my old colleagues, and Gower St. is Technically not part of the TLRN - it's a Red Route side road.

Although TfL are the Traffic Authority for the bits with red lines, they are not the Highway Authority.

Reporting on the aboeve link won't do any harm though, it'll get pushed through to the relevant Highway Authority anyway...