Finding the private key is going to be rather difficult, especially if you might even have used a web based wallet (such as coinbase, or blockchain.info).
To start with, I'd immediately set up a new wallet, and create a good backup of it. Store the backup in a safe place where you know you can find it in the future if necessary, and put a big label on it that says "My bitcoin mining wallet!"
Then change the payout address to this new wallet right away. That way even if you don't find your original wallet, you won't continue losing more bitcoins to the unknown address.
Given how new and uninformed you were, it is even possible that you accidentally put someone else's bitcoin address as the payout address for the mining. You wouldn't be the first newbie (or second, or third, etc.) to make that mistake. I've responded to many newbies who come here to bitcointalk.org and their first post is:
I misunderstood and accidentally used my "sending address" from my wallet instead of my "receiving address" when I tried to send bitcoins to my wallet. How to I recover the bitcoins?
Bitcoin doesn't have a "sending address". In every one of these instances, what the newbie accidentally did was choose from their transaction history an address that they previously sent to. In other words, someone else's address.
If you made this same mistake when you were a newbie setting this all up, then you'd have to figure out whose address it was, and ask them to send the bitcoins back to you.
If the address belonged to an entity that no longer exists (such as silk road, bitfloor, MtGox, etc.), then getting those bitcoins back would be extremely unlikely.
