Puke-blocked drain in flat above caused leak down into my living room, damaging electronic equipment.

Living in rented accommodation, 3 flats in the building, we're underneath the other 2. Oh, and I'm in the UK (England). First off, the flat upstairs has been warned about blocking up their drains before, as we received an email to all 3 flats warning everybody of it, someone had to come out and unclog it etc. A few weeks ago, we woke up to find that yellowish water had leaked through our ceiling, down the walls, and waterlogged my digital piano (worth about £700-800) Luckily it was unplugged but could have caused a fire. It's the most expensive thing I own and I'm pretty heartbroken about it :( Called the landlord, their maintenance guys discovered that one of the bathrooms in the flat upstairs was waterlogged, and the drain was blocked with vomit. Lovely. 4 weeks later now and we've been told that they're refusing to accept liability for the damage to my piano. So I'm looking for a little advice regarding where to go from here? Regarding insurance, I was initially told that the other flat would have to make a claim on theirs, but since they're refusing to accept liability it seems this is no longer the case. It seems to me like someone has to accept liability, it's definitely not me, and if they're refusing then is it the landlord? (Further info about insurance and stuff: It's in the tenancy agreement that we have tenant's liability insurance, and I've called our insurance provider and been told that a possibility is that I make a claim, and then they recover the costs from the other side, as long as they have insurance too. Which they should do as it's most likely in their tenancy agreement too. I just don't see why it isn't as simple as, they caused the blockage, they're liable for the damage (ESPECIALLY as they've been warned before about it.) Oh and the landlord is useless (well, the letting agent acting on the landlord's behalf), as in they take ages to respond to anything and it's taken numerous chase-ups to simply find out this much. I've also taken lots of pictures of the damage btw.) Help??