Is your performance today(when working ) ok?
Inability to power up is not likely to be caused by the case.
On occasion, the front power connection fails.
You can test by momentarily shorting the motherboard front panel PWR pins.
You start with the processor you want, only then can you start on motherboard selection.
If contemplating a new build, start with the performance of your I7-4790K.
It has 8 processing threads and a passmark rating about 8057. That is when all 8 threads are 100% in use.
The single thread rating is 2470. Single thread performance is probably more important for everyday quickness.
For example a 12th gen I3-12100 also has 8 threads but scores 14540/3550.
For your described use, that should be plenty.
I3/5/7 no longer indicates hyperthreading and core count. It is more of a relative power positioning today.
If you have more than 6 devices, you are probably looking at a Z690 based motherboard like this:
https://www.newegg.com/p/N82E16813162039?quicklink=true
That motherboard uses DDR4 ram.
That is good, DDR4 is cheaper than DDR5 and any performance differences are minimal.
Your psu is likely ok as is the case.
One thing I would put in the budget is a larger high resolution monitor.
Keep the old monitor as a side monitor.