You need to check those joists out as well- try stabbing them with a flathead screwdriver, if it sinks in more than about 10mm then you need to have a good look at what they're supporting and how feasable it is to replace or reinforce
EDIT Just reread and you're aware of the joists already, good
First, have a look in the loft and make sure that wall isn't carrying any substantial load (some stud walls are loadbearing). If it is (there could be a purlin block or something on top of it) then it'll need propping.....
You need to cut that plate out and replace it. How deep is the skirting board in the bedroom- if it's more than the thickness of that timber then you're in with a chance....
Remove skirting board from bedroom wall
Cut through the plasterboard (padsaw is neatest, angle grinder with a disc will do) from the bedroom side- aim is to cut above the sole plate and below the top of the skirting. Make sure you cut the plasterboard all the way across including where it covers the vertical studs and vertically at the corners
Now cut through the studs about 5mm above the sole plate (use a hacksaw, there'll be nails or screws)
Remove the sole plate- it looks as if it will fall to bits but you may have to cut it into bits to get it out. Ideally you want the old nails out of the joists- if you can't get them out then saw them off flush
Sort the joists out
Put a new sole plate in (looks like 4 x 2, whatever you get make sure it is tanalised), fix it to the joists (big screws and an impact driver will be best, big hammer and nails probably equals bye bye plaster on the ceiling below)
Pack the gaps between studs and new plate with window shims
Skew screw the studs to the sole plate (pilot drill the studs to set the angle you want). If you can't get a good fix then screw blocks of 4 x 2 to the sole plate next to the studs then screw through the studs into the blocks
Stuff the gap with rockwool
Put the skirting board back on in the bedroom (you can replace the bit of plasterboard if you want but it isn't compulsory, you'll need to pack it out with something so it sits vertically)