This is suchhhh a late reply, but here are a few that come to mind. You might have already seen some of these but here they are anyways... lol
Just Between Lovers (Rain or Shine) -This has a scene where the FL gets a cold and the ML comes through her window and takes care of her.. super cute.
A Female Student Arrives at the Imperial College - This one is a historical, I don't know if you're into those, but in the first few episodes the FL gets poisoned, and the ML is worried about her, etc..
Assistant of Superstar - I don't know if this one quite fits the list, but the FL is in pain (period pains) and the ML rushes over to help take care of her. Because he assumes she's sick. (This scene doesn't last long) Butttt he helps her overcome her fear throughout this drama, so it's worth the watch anyway. haha
Dear Little Mermaid - This is the definition of having a ML take care of a sick FL. The FL is sick with a disease, so she's in and out of hospitals. (don't worry, it's not cancer or anything, and she doesn't die from it. It's not like a horrible sad illness we see in other dramas, she pretty much just ignores it) She tries to live a normal life, and falls for the ML. Soon the ML finds out about her sickness and decides to take care of her. (Not only is this drama funny and romantic, it has a sweet and caring ML who wants the best for the FL.) Definitely recommend if you have not seen it.
Falling into Your Smile - This one you might have seen, but not only does the FL take care of the ML when he's sick, the ML also takes care of the FL despite him not feeling well. It's super cute and he always takes care of her. Especially during her injury.