From what you are saying here the power button is not depressing when you press it. Thats a sign of being gummed up.
Lets try this: with the systems power cord disconnected, get some household spray cleaner dampen a paper towel and wipe the button area, pressing it on & off so the cleaner can work its way in, don't flood it! As you don't want the cleaner to drip on the insides. You may need to apply a clean towel to catch the dirt.
Did that help? It not try some isopropyl alcohol (85% or better) this time. Again, using a paper towel this time at the end try spraying the button cutout area with a short blast of can'ed air.
Any better? The last thing here is to try a little (and I do mean LITTLE) lube. Take a cotton swab and spray some WD-40 onto it so its just slightly damp. Again, wipe the button cutout pressing the button on and off so the lube can work its way in.
Let us know how it goes!