Upper Respiratory Infection...can my cat catch it twice in a month?

Hi, I'm a new cat owner. I adopted Biscuit from a shelter about a month ago. She developed a really bad UTI just 2 days after being at home. She was put on antibiotics and nebulizer treatments. She didn't eat for a week but I feed her with a syringe and magically on day 8 she started eating again and by day 12 she was back to normal.
I really wanted a 2nd cat to keep her company and I also felt so bad for all the kitties at the shelter that I needed to take more than just one. Well, I adopted a 4 month old kitty (Muffin) that was recovering from tapeworms and being spayed. I isolated the kitten from Biscuit until I could get a the green light from the vet. Yesterday the kitten was checked by the vet, the tapeworms are gone and even though she is underweight, the vet said she's healthy. Today the kitten started coughing and I'm so scared that she's developing a respiratory infection. I also wonder if Biscuit could get sick again. The kitten was exploring the house yesterday and Biscuit was smelling her and other than some hissing. They're getting alone great. Should I isolate the kitten? Can Biscuit really get this cold again? I started giving both cats Lysine and nutrical. Any other advice?
Thanks!

The lysine should help quite a bit! I'd consider adding 1/2 a capsule of Jarrow lactoferrin to the food. That also helps combat viral infection, and works synergistically with the lysine. Works for mine! Though I give my 10 pound or more adult cats a full capsule once a day.