It's probably not Perspex.
If it's ABS, almost nothing will stick to it or stick it together. If it's polystyrene (unexpanded), pipe cement from a plumbers merchants works (I've mended flaps,lids, etc. from our fridge that way). You'll get a better repair that way than trying to bond it to wood. You can clean up the results by carefully sanding down the glue to be level with the original material (let it set hard then go down through the paper grades), then polishing with Brasso to a fine shine.
If you really must, roughen the surface with coarse sandpaper and use a flexible version of Araldite. The bond won't be strong though, as it's only really mechanical rather than chemical.
For proper Perspex there is a very expensive glue, available from sheet stock suppliers. The traditional version was Perspex chips dissolved in Chloroform, but it has a limited shelf life and is not recommended these days.
Plan B: can you replace the whole thing in wood - it would probably last longer and be a better insulator.
E.