AP is less than 10 minutes walk from the railway station with regular service (every 5-10 minutes at peak time) into Finsbury Park and on to Moorgate, or take W3 bus to Wood Green or Finsbury Park.
Nice views, good ice.
Evening patch tends to be busier than morning, morning CAN get a bit intense if the elite ice dancers are in.
But I think a well managed rink.
In theory you are meant to have a patch pass signed by a coach (doesn't necessarily have to be one you are having lessons with). In practice they don't seem to check but I think you're best off following Black's advice.
I will PM with what I know about coaches.
Sobell has no patch as such, just public and group lessons you need to sign up for in blocks. Good if you like group lessons though the general level is more like NISA Bronze/Silver/Gold. It is small.
Lee Valley there are buses from the rink into the City but they are a bit slow, or walk 15 minutes to Clapton station which is less than 15 minutes from Liverpool Street. Same size as AP. People I know who skate there like it.