Personally, I've never had experience with that before. I'd take the easy way out and just get a new drive. You can normally find one on ebay for around $20-30.
To make it easier on yourself, considering I don't think you'd have experience flashing a drive, make sure you get the same drive that's currently in the 360 and when you replace it, take the drive apart and swap the drive board (the circuit board in the drive) on the old one in place of the one that's in the new one.
This will give you a new drive while keeping the DVD key on it so that you don't have to flash it.
If you're still having problems after doing that, though, then you'll need to put the boards back in their respective drives and copy the DVD Key off of the old drive and flash it onto the new drive so that you'll be able to play 360 games still.