After a while of use the front cover of the PC can come loose causing the issue you are describing. What I would do is turn off the computer (actually unplug it to ensure you don't get shocked) and remove the front-panel of the computer (usually accomplished by prying it off starting at the bottom). If you think you will break the case doing this, remove the side-panel as shown here
, then check to see if the front-panel is secured with screws. If screws are securing the front-panel, remove them. Afterward try prying the front-cover off again.
CAUTION: Touch the metal chassis of the case before placing your fingers near electronic components inside the computer! If you don't, even the slightest static-electricity discharge can break electronic components inside your computer.
Just so you know, I've removed front panels from computers multiple times. Never have I removed one without some form of cracking/popping noise being generated. Therefore don't get frightened if you hear one, it's quite normal (especially on Dells).
After you have removed the front-panel of the computer, inspect the power-button-depressor. We're checking to make sure it's not broken or that it slipped out of its housing. If the plastic power-button-depressor is broken, check to see if you can position it in a way that it will still be functional. If not I would remove (break off) the power-button-depressor, install the front-panel making sure that it seats firmly, then use some form of a small device (such as a toothpick, pencil (use the flat end), or some other small device) which will depress the actual power-button inside the front-panel.
However, if the power-button-depressor is not broken, simply install the front-cover making sure it reseats firmly. Afterward check to see if the power-button-depressor functions correctly.
If you're still encountering the same problem I would just break off the plastic power-button-depressor (requiring you to remove the front-panel again. Once again, don't forget to touch the metal chassis to prevent an inadvertent static-electricity discharge) and use a small device (as mentioned above, a toothpick, pencil, or other small device) which will easily depress the power-button inside the front-panel.
Hopefully reseating the front-panel will resolve your issue. Yet if it doesn't, you can still operate the power-button by use of a small device. No, you will not be shocked, nor will you come close to exposed electrical wiring (all wiring inside computers is insulated allowing you to touch it while the computer is running). Therefore the use of a small device to power on your computer is quite practical if your original power-button-depressor broke.
Let us know how things turn out.