A flaky on/off button or power button board are my first thoughts, the on/off button you can inspect and perhaps spray with an electrical circuit cleaning type aerosol but unless you are a qualified electrician or TV repair specialist I would not recommend you open up the monitor yourself, they can hold enough residual charge to do you serious harm and worse case scenario even cause death, not being dramatic here either.
The alternative is to have a suitably qualified person inspect the monitor but you would most likely find that you could purchase a new screen for what this would cost.
Have you tried the Dell diagnostics, see below;
Monitor Specific Troubleshooting
Self-Test Feature Check (STFC)
Your monitor provides a self-test feature that allows you to check if your monitor is functioning properly. If your monitor and computer are properly connected but the monitor screen remains dark, run the monitor self-test by performing the following steps:
Turn off both your computer and the monitor.
Unplug the video cable from the back of the computer. To ensure proper Self-Test operation, remove Analog (blue connector) cable from the back of computer.
Turn on the monitor.
The floating 'Dell - self-test Feature Check' dialog box should appear on-screen (against a black background) if the monitor cannot sense a video signal and is working correctly. While in self-test mode, the power LED remains green and the self-test pattern will scroll through the screen continually.
This box also appears during normal system operation if the video cable becomes disconnected or damaged.
Turn off your monitor and reconnect the video cable; then turn on both your computer and the monitor.
If your monitor screen remains blank after you use the previous procedure, check your video controller and computer system; your monitor is functioning properly.