Hate to chuck a spanner in but?
If you have a close look at the hole that is to be filled, the feed pipework has at least one in line stop valve, generally called a ball O fix It is a small shut off valve that would allow you to stop the flow of water to the tap if you need to change a tap washer.
if you block up this hole then access to the valve will be denied, indeed both the hot and cold feed pipes will in all probability have such devices fitted, just cant see the second pipe.
As fix, how about using a piece of plywood cut to slip over the hole you could then use silicone [which is a good glue] to set the ply in place, hole blocked, no more mouse runs from that source.
As an aside are you sure that the little furry ones are accessing the house via this hole? I can see no indications of any fraying of the edges of the plasterboard to indicate this is a mouse highway, normally if mice are using this as access I would expect to see rub marks, minor scratch marks and similar to indicate a mouse run?
Ken.