I’m going on my 1st European adventure in just over 5 weeks.
I’m not staying at the Clink, but my hotel is within 5 minutes walking distance from there.
I will be catching the underground train from Heathrow airport straight through to Kings Cross St pancras station. Apparently it’s the Piccidilly line. if you look on google maps, u can get walking directions (the Clink looks pretty easy to get to from the station) and u can even go into the street view mode and do a practice walk online.
Hope this helps, good luck and hope you have a ball on your trip