My iMacs Display Screen Fell down
You have two distinct problems with your iMac that you’re unfortunately conflating with the most obvious damage - the broken display.  


The display is most likely broken and will need to be replaced.  I can be tested, but not in a typical home or office environment.
The cable connectors on the logic board may be damaged as a fall like that will literally rip them off the board.  A visual inspection should be enough to verify if they are good or not
The fan spin is likely because you replaced your hard disk with a non-Apple one that doesn’t have a temp sensor.  A missing sensor will cause the fans to spin at full speed
The slow performance is likely due to the erroneous temp readings (from the missing sensor) and the Mac thermal throttling in response.


To fix the fan and speed issue, you simply need an inline temp sensor to send the correct data to the SMC.  However, the screen is quite expensive and you should get a quote on a replacement prior to doing this - it may be much more cost effective purchase a replacement iMac of the same vintage and sell that one for “parts.”

Also, if you’re going to replace a HDD and go through all the trouble on an iMac, you should spring for an SSD.  The performance gains are well worth it.