If I buy the same game I already have downloaded on my computer from a different steam account, will I have to re-download it?

A while back, I got completely locked away from my old steam account because of a hacker that bothered me more than a couple times. He completely changed all the information on the account so I can’t get it back. I was going to re-purchase a game that was already downloaded on my computer from my last account (I can’t play it since it’s linked to the other account). My question is, if I purchase it again on another steam account, will I need to redownload it or will I be able to use the already downloaded file to play?

No, you won't have to re-download the game files. From Steam's support page on Steam Account Use :

Can I use multiple Steam accounts on one computer?
Yes, you can access different Steam accounts from one computer. However, you may only access one account at a time.

The accounts will share the core files for any games the accounts have in common, but each account will maintain its own custom content and configuration files.

To answer the original problem, Steam does not use any of your personal information to restore control of your account. When I had to regain control of my Steam account, they asked for one of the Credit Card number used to purchase the games in the account.

Contact steam and retrieve your account.