That's right. I forgot, you only have the patella problem and not a TKR, too. I did have a knee replacement, and most people on the knee part of the forum either have one or are waiting for one. Despite you not being in that category, you're still in a good place to get commiseration and advice on your problem.
I strongly suspect my knee cap cracked because it was shaved during my knee replacement to make it thinner so my surgeon could apply a patellar button on it. The good news is when it cracked, I strongly suspect the patellar button supported it enough so it didnt' break completely.
X-rays confirmed the crack, so my surgeon put me in a brace and had me come in every couple weeks for a month and he took x-rays to monitor it. If he saw the crack widening, he was going to have to do surgery (which I suspect would have been the same as you had). He made it very clear I needed to baby the knee as much as possible, so I did. I wore the brace from the end of January until the end of March. Then I had strict instructions not to do any squatting on the knee or deep bending for three months. The knee has been fine since then.
I'm not sure what the expected recovery time would be for your situation, but I suspect it might be longer, because the bone has so much more knitting to do, and you had the surgery, which wasn't trivial.
I don't remember if you said, are you resting, elevating, and icing (with a thin cloth between the ice and your knee to protect the skin)? That can help the swelling and pain. The stinging could be nerves reconnecting. That's fun, but it should pass after a month or two (some people have gotten a prescription to help with that, if that's what your burning sensation is caused by). In addition to the pain, you may be experiencing numbness on the outside of the knee area too, depending upon the type of incision they did to repair your kneecap.