I cancelled my Xbox Live account many years ago, and sold the Xbox 360, but I still have the hard drive. If I put that hard drive into a new Xbox 360, and make a new profile, will I still be able to access all the DLCs that I purchased on the previous profile?
Short answer: No.
When you purchase anything on the Xbox marketplace, you are purchasing it for your account. Even if you did not have the DLC on the hard drive, you may freely re-download them, providing your signed in to the same account.
That said, you state that you intend to set up a new profile. Your new profile would not have these DLCs "authenticated" to it, so they would not work.
Your best bet is to simply use the old Xbox profile.