That's exactly what Fish! is for, but again, don't bother.
You could consider Fenchurch in the Sky Garden, on top of the Walkie Talkie building... it's expensive, but probably within your budget, and you get a nice trip to the Sky Garden also, which is worth it. Booking essential. When I went to the Sky Pod Bar... they took a table deposit (£25 each, card details only to secure) at booking, encase of a no show, which considering the location, and that they don't take walk-ins, I thought was fair enough.
I've not eaten there personally (yet), but the food is excellent by all accounts.
http://skygarden.london/fenchurch
** Checking the menu, it's not exclusively a fish restaurant as I thought it was (or was at opening), but it has some nice fish dishes.