Hi all, I hope someone can help me!

I moved into my flat (rented) approx 10 months ago. When I moved in, there was a problem with one of the toilets, and we informed the estate agents immediately. They called their own plumber to come and rectify the problem.
Now, 10 months on, I have received a water bill for over £500, as the £36 per month I pay the water company won't cover out new meter reading.
I rung the water company, who said it must definitely be a leak. After determining the leak was on the property, I switched off the water temporarily. Whilst investigating, we noticed the cistern from the broken toilet had completely drained. We have had a plumber acquaintance come out and have a look, and he said that the work originally done to the toilet has led the water to slowly drain from the cistern into the toilet, meaning that the cistern is constantly refilling itself. Due to the silent but steady nature of this problem, it has gone completely undetected.
I spoke to the estate agent this morning, who was quick to deny any fault and refused to absorb any portion of the bill.

Am I being stupid/naive to say that this problem was not something I should have to pay for completely by myself, and that the estate agents/landlord should at least offer to absorb some of the costs for the shoddy work that was initially done by their contracted plumber?

Any help/opinions would be appreciated!