Poor you - and poor dog!
If he/she is following you, a stranger, round when you are in the house, the animal is desperately distressed and worried that is has been abandoned.
Your colleague is a selfish bitch.
I'm funny about sleeping in someone else's bed at the best of times, but living in this degree of filth would make me ill. The house must stink.
You say your landlord lives in the flat above yours - how approachable is s/he?
Would they let you bring the dog home for a finite time (the rest of the fortnight) if you explained the circumstances and promised to ensure that your home was thoroughly cleaned afterwards? It's not like you are getting yourself a dog which could be there for 15 years.
There have got to be health hazards involved, too. If you can't find another solution, and you have to stay there (poor dog can'r just be abandoned - it needs company as well as feeding/walks) then could you clean and effectively live in the bedroom - treat it as a bedsit - take the kettle in there, watch TV on your laptop and just give that poor animal a bit of TLC. (Yes - I admit it - pets and kids are very high on my list of priorities, and I would personally put up with almost any discomfort to ensure they were okay. They don't get a say in where they go or what happens to them - it's awful for them).
Does she have a working washing machine/dryer - wash her sheets etc every day that you are there. Also - take photos of the squalor. If she complains about anything you've done to make like more comfortable for yourself, you will need proof of the state of the house.
This is horrible for you.