The bad news is that finding an OE unicouple handle isn't very easy; likely you'll have to find a "parts" dishwasher to scavenge one. For some reason, new unicouple assemblies cost a lot.

The good news is that only the handle on your unicouple seems broken. I can't tell without more pictures, hint, hint. AFAIK, the handle could be cut from the unicouple and then be replaced by another piece of plastic or metal. Won't be terribly pretty but that'll do until you find a replacement. You might try eBay for the unicouple assembly. They're rare, but they do show up from time to time.

As far as the connection to your sink goes, you can buy adapters, new style and old style for your faucet at almost any decent hardware store.

Good luck.