Well that's a bit of a relief! Hope he gets better soon. You might want to get some L-Lysine
to help give his immune system a boost. The usual dose for a URI is a 500mg capsule per day (split between 2 meals). You can mix it in with canned food.
As for the antibiotics, make sure to watch out for any signs of inappetance. Some cats (like mine!) get digestive issues from them and the last thing you need is for Jake to stop eating again and be faced with more fatty liver problems. Call your vet right away if Jake has any diarrhea or vomitting or just loses his appetite.
Good luck, and let me know how he does on the Wellness.