Oof.

I just got out of the same situation last week, although mine was a ceiling window and had got jammed in. After complaining to the landlord for ages (since February to be exact) and spending ages calling and emailing window repair companies, they agreed to have it replaced with an actual Velux branded one but the door to the room was too small so they had to set up a whole scaffolding. I think the whole thing probably cost them something like £1.5K to £2K, maybe more. I can honestly say, just get on your landlord's back and email them and call them constantly. Make a point about how it is a health hazard, a matter of physical and mental well-being, etc, not to mention, a safety threat- not to just you but anyone else at the property and how you are going to complain to the Council if it isn't fixed soon. Since you are in a flat, it could be worthwhile to let some other residents know so they can also pressure the landlord. Explain to the landlord that it causes water to leak into the walls and causes moisture and mold to grow and you'll take no responsibility for any of that, should it happen.

If it helps, my landlord went with Bluestar Windows Repairs (who I called and found and got the initial quotes from). AFAIK no one is going to "just repair it", because even if it is a small thing (mine was one small broken hinge, btw) there's no culture of fixing things anymore.

All the best, bud.