Agreed. I would send the estate agents a very short letter stating what you asked for, what was delivered and what you've had to get remedied yourself. Point out what you owe, point out what you've had to spend, and then point out that miraculously both sums cancel each other out (even if they don't, make them by reducing the costs on your side). Remind them that you have no contract with the contractor, and if the contractor feels they do then ask them to supply a signed copy of it. Clearly they can't, so I'd ignore it.
You may be out of pocket, but I'd use common sense and walk away at this point.
Also, don't forget to get a legionella risk assessment done for the property if you're renting it out, a lot of people forget about this. Apologies if you're aware and have already sorted it.