Cat bites are so difficult because puncture wound!
Unsolicited advice spoiler space...
I find it helpful to mix warm salt water, about as salty as tears, in a cup, and soak the infected finger while I'm watching TV or surfing Mastodon or whatever. Or maybe half an hour, once or twice a day
I've also been known to do this with rubbing alcohol straight from the bottle, although it didn't seem to work better than the salt
For keeping it clean while working, I cut a finger off a nitrile glove
(And obviously go to the doctor if it's getting worse instead of better, over the next couple days)
Did not expect to be crying into my old guy's fur about another cat!
Stupid shelter, stupid adorable cats making me fall in love with them. ๐
I'm so sorry honey. It's hard when you have a loving heart in this world!