Originally Posted by Demise

What you're looking for is called "Cancellation of Removal".

The requirements are basically as follows:
1. You need to have been here for 10+ years prior to being placed into removal proceedings. (With the exception that an NTA with "Date to be set", "Place to be set" does not stop the clock).
2. Good moral character for the preceding 10 years.
3. Not convinced of any CIMT, certain other felonies.
4. Have a USC or LPR, Spouse, Child(ren), or/and parent(s) who'd suffer "exceptional and extremely unusual hardship" if you were removed from United States.
5. Have not been granted cancellation of removal, suspension of deportation, or relief under former INA 212(c) previously.
6. Have not entered US as an alien crewman (C and D visas) or entered on a J-visa subject to the foreign residency requirement.

This is a hard process, it is limited to only 4,000 grants per year, and there are way more applicants than slots. It can only be sought in removal proceedings and ending up there isn't exactly easy when you want to end up there. On the up-side you are eligible for an EAD while your case is pending but it's case that can only end in one of three ways:
1. You win and get a green card.
2. You lose and get a removal order.
3. Something else comes along and you could withdraw the application and adjust instead.

Can you win it with a sick child? I'd say it's definitely possible but not easy. I am a person that's very DIY, I think any idiot who can read, can do their own AOS, I think you can do your own waivers and VAWA cases if you know what you're doing. For COR my only advice is get a good lawyer.