I know generally it's wise to finish the sleeve with hand tattoo but... 🙂
I wanted tattoos for long time and decided to go first with the simple ones I guess, like I remember wanting a dice tattoo on my hand thats it, I also wanted to cover the scars on it.
Things happen. Hassling the artist isn't going to make him or her in a huge hurry to get back to you. Just leave it. Let them get well. There are lots of bugs going around at the moment.
Also, they did YOU a favor by not making you sick. So calm down.