If I sent a lightning network transactions off-chain and it is confirmed, what is there to stop me sending the on-chain transaction to another wallet I own. When the lightning network channel is closed, then it is no longer available?
The funds you would be attempting to spend are actually in an on-chain transaction. They are locked up in a 2-of-2 multisig on a confirmed on-chain transaction. You cannot move those funds elsewhere on chain without the other party signing a transaction to move the funds. That signing is done with the commitment transactions (which update each person's balance) and at channel closing.