Station ?

Nothing to do with railways. There's a connection with my earlier entry, guessed by trewilliam (and probably Adrian).

are we inside king's college off strand lane ?

Ah, you're thinking of Surrey Steps, next to the 'Roman' bath, I think? You're getting closer - need to continue east a bit.
Strand Lane isn't open on Sundays (as you say it's part of KC) but you can walk up and look through the window at the bath on weekdays.

no, I knew it wasn't surrey steps, but seem to remember that strand lane used to continue up to the strand once. you used to get the bath key from over the road when the n.t. still thought it was roman. just fishing really; looks too modern to be under somerset house, and the modern bits down by where the barge is kept are much slicker. could it be leading up to the adelphi ? though I can't visualise any such modern handrail on the rather more forgotten looking steps there. . .

Not Somerset House or the Adelphi - you're going in the wrong direction. These steps would have led to the river before the Embankment was built...

could be milford lane behind the astors' house on temple place, now open to the public for exhibitions from time to time.

Hooray, yes, Milford Lane. Actually the steps are the Essex watergate that leads up to Essex St but since they lead up from Milford Lane your guess is correct!
2 Temple Place is a great house to visit if you get the chance. I was there a couple of years ago for the Open House after party.

I caught the pre-raphaelite exhibition there a couple of years ago and snatched some interior shots . Enjoyed the less commercialised open house in the early days more than now; too many self-publicists taken over, though I still go out of course. The out of town weekend is often more fun, with the old sense of amateur enthusiasm intact .