Separate laminate from chipboard
It may be a hot glue used, try a wet / damp towel and a hot iron to loosen it, but you will need to do so while it is extremely hot. Be careful. A broad putty knife may be handy for this.

Picture courtesy Lowes
This may not be an easy solution, while hot the laminate will be very weak and flimsy, let alone the hot glue may want to bond the laminate to the knife as you are going, if it does anything at all. Get it too hot, it will damage the laminate too.
If you really feel you need to cover the hole, recut the hole to a square opening, then cut the filler piece to match the size of the hole. Another tip would be to taper the sides of the new cut and the filler so it will stop at a given place.