Bed-bug infestation - who should foot the bill?

Other

I bought a flat in February and did a light refurb over several weeks (I did it myself with just weekends to work on it) which also involving replacing the carpets.

A tenant moved in about 6 weeks ago, and brought in an old bed for herself, a new cabin bed for her daughter and an old mattress for her daughter's bed that had been 'in storage at a friends house'.

2 weeks ago I got a call from the tenant saying that they were being bitten at night, and demanding that I do something about it.

I explained that the previous tenants did not have pets, that the old carpets were thrown out and the floors all swept and vacuumed before new underlay and carpet was put down, so I discounted the possibility of fleas.

I got her to catch one of the insects she believed to be the problem, and subsequently found it to be an adult bed-bug.

I've looked into getting the place fumigated and found that they can be brought in in furniture, bedding, clothing, even by visitors!

The cost of getting the place fumigated is likely to be about £200 and what I want to know is:
a) Who should foot the bill? I'm certain that the bugs came in with her furniture (probably the mattresses) but can't prove it.
b) Can the tenant withhold rent until the problem is sorted out?

Many thanks for your help.
Paul.