By not turning on, do you mean the screen is black and there are no lights indicating whether the power is on, the HDD is being searched?
Do you get a flash screen with the HP logo?
When I diagnose a laptop for refurbishing the first thing I check is whether the flash screen comes on. If it does, I press the key they show to get into the ROM-BIOS and then check the configuration. You may also have diagnostic tools to check memory and the HDD.
If the computer shows lights and it sounds like something is happening, you may have a bad screen, connect an outside monitor and see if that works. Look up how to replace/repair the screen on the Internet.
Another reason for nothing to happen when you turn on, maybe static electricity on the start button. Take out the battery, disconnect the power and press the start button for about 15 seconds. Plug everything on and try to start.
If the memory is bad, computers will not start. Remove the memory, clean and try installing one module at the time trying to start every time. Unfortunately you would have to install good memory to check this diagnostic. Do you have old memory that would operate in this computer?
If the computer boots up but you get a message that media is not connected, you have a failing HDD. You could try reinstalling your operating system. If you can’t, then it is time to get a new one. Get a SSD instead, your computer will fly.
Finally, take out the battery and see if the computer starts. I have had this happens when the battery is bad or is shorting out.
Chances are that one of these things will give you some information why your computer doesn't start.