Various unsorted thoughts:
Add pedestrian crossings to Kirk Brae, Blackford Glen Road and Mayfield Road.
Extend ASL on Mayfield Road ~3m forward, so that cyclists start on the flat. Advanced green lights for cyclists.
Painted ("car", not cycle) lanes through the junction. A cycle lane from Liberton Brae to Mayfield Road should be considered.
Allow left turns for cyclists from Kirk Brae and Mayfield Road.
Single yellow lines on Mayfield Road should apply till at least 9:30am (currently 9:15am).
Braefoot Terrace should have double yellow lines, single yellow kerb flashes. No loading should apply till at least 9:30am.
Liberton Brae should have double yellow lines and single yellow kerb flashes northbound, at least as far as Alnwickhill Road. Southbound, there should be at least single yellow lines and kerb flashes - ideally double yellows.
I'm not sure what I'd do with that traffic island on Mayfield Road. Probably get rid of it. It can't move any further north, because of the junction, and if it's moved far enough south to fix the problem for people turning into KB, it's not going to be on the desire line for pedestrians crossing the road. It creates a nasty pinch point for southbound cyclists too, regardless of its position. A proper pedestrian crossing could be the answer?
Currently, if you are only able to filter to the end of the southbound bike lane on Mayfield Road, you almost certainly don't get through the junction at the next green light. The bike lane should continue past the pub, at least.
Ban motorised traffic going from Mayfield Road to Kirk Brae? Too radical?