The reason I asked was that I'm not quite sure what she would do with the cheque.

If she presents it to her bank, they would tell her that they have no relationship with your bank but they could pursue it (for a big charge). Their bank would then (either electronically or by telephone) get in charge with your bank for payment.

Your bank would likely say "huh? Euros? Ok, for a fee" and pay out, electronically.

Eventually I'm sure she would get her money but it will probably involve fees for both her and you since the banking system is not really set up to resolve these sort of transactions.

Don't know, but maybe wiring the money directly is cheaper and faster.