While drilling into kitchen wall, hit a wire - any advice?
Obviously, safety considerations like isolating that circuit are paramount, and note that even if that circuit's breaker is off, then the neutral touching earth can still trip the Earth Leakage Circuit Breaker...

Find the socket / box that that wire goes to, disconnect it, then pull it out from the other end (which you need to find - either in the roof space or under the upstairs floor) then replace it.

If the copper has been damaged then just covering over with tape may not be safe.

Looking at the plastic cover, this looks like you are in the UK. You can hope that the wire is not totally trapped somewhere...