If your old laptop has VGA out you could use that. You may need a converter depending on what inputs the old TV has.
You could also use a USB video adapter to send video to the TV. That might be a bit tricky since you may have to install a driver for the adapter and doing that without a working display may not be possible. You could borrow an old monitor to get it set up,
You will find that the edges of the desktop will be overscanned. Some TVs let you turn that off so a PC will work better.