Since you've already tried reinstalling the OS X, there are two causes for this problem. Either the partition structure of the hard drive is corrupted or the hard drive itself is failing which is very likely to happen as you mentioned it had been slowing down before. The fact you are able to boot into OS X Utilities screen indicates that all other core components of the Mac are fine.
In rare cases, corruption in user account can also cause this. So lets rule that out first.
We can create a new account and see if you are able to login to that account.
• Turn off/shut down your Mac.
• Turn it on and immediately hold down the Command(Apple) and S keys together for 10 seconds to boot into Single User Mode
• When you see the text prompt, type in the following terminal commands to create a brand new Administrator account (hitting return after each line). Also, note down the spaces between the commands, like a space between mount and -, space between -uw and / etc
mount -uw /
shutdown -h now
(If it returns any errors after any command above, then you are entering the commands incorrectly. Make sure you enter it correctly)
When you enter last command and press return, the Mac will shut down. So turn it back on, you should be able to create a brand new Administrator account (when creating the account, if you're asked to transfer information, select Don't transfer)
Login to new account. Are you able to login to new account ? This will tell us if the issue is related to user account.