Legal advice on payment of Recruitment agency fees for candidate who lied @ interview

Description of legal work: We employed an Agency to recruit a service engineer.
They provided a candidate which we interviewed.
At interview, candidates complete a form stating how much experience they have had with different systems, what training they have had and how they rate their overall competence.
The candidate concerned filled the form in inaccurately as a result of which we made him a job offer.
He accepted and started working for us.
After several weeks it started to become clear that his competence did not match the criteria he had put on the form.
At 6-weeks he attended a Probationary review at which time he carried out a practical test designed to directly assess his ability. He failed the test abysmally and we have now served him notice.
He is entitled to 1 week's notice but we have given him 1 month to give time to find an alternative engineer.
We have not paid the Recruitment Agency and they are now pressing for their money.
According to their terms of engagement (which we signed) we are entited to a 10% rebate due to the length of time that has elapsed since we took him on.
We do not feel inclined to pay the Agency anything unless they can provide an alternative candidate. Their website says "Before you see any candidate, we will have interviewed them thoroughly, confirmed they meet your criteria..."
A copy of the Agent engagement letter is attached together with their Terms and Conditions (with names and contact details redacted)
The issue is very unlikely to go to court as the Agency would no doubt like to continue working for us but we require assistance to know how strong our position is.
Extra notes: Mandatory requirements to bid:
1) In depth knowledge of relevant UK law
2) Fluent communicator in English