My instincts are:

• Your SIL is in debt
• Your brother doesn’t know
• She’s pin a tryst because she can’t pass affordability checks with a loan over her head
• So she wants £10k from you

Alternatively:

• Your SIL is genuine
• The bank and solicitors will ask about the source of funds
• You will sign a letter saying it’s a gift
• Either:
• a) it isn’t a gift, it’s repaid, which means your SIL and brother have committed mortgage fraud
• b) your SIL decides she doesn’t want to repay, and you have no legal instrument to recover the money

Change your mind. Say you don’t have the money. Say you thought you had more than you really did. Say something came up or that you think you might be made redundant. Tell the truth, lie, prevaricate, whatever you do do not secretly lend someone £10,000 as a putative deposit gift so they can defraud their mortgage provider.