He is not the nurturing, caregiving type. By choice as well as by nature. He is the Joker, Gotham's Clown Prince of Crime, he's not some... nursemaid fussing around after his underlings.
But then, you aren't just one of his underlings, are you?
He'd ignored it at first. You'd kept doing your job, like you always did, and if you were a bit pale? If you were a bit slower at things than normal? If you interrupted him with the occasional cough? He could ignore it. He'd hardly noticed you eating less at first - this wasn't exactly a lifestyle that led to regular 'sitting down to meals together', it wasn't his job to make sure you put food in you (even if you always made sure he ate). He'd ignored it, and you'd ignored it, and nothing was the matter.
Until you'd collapsed. Until you'd been too weak and disoriented to get back up. Until he'd practically had to drag you to a couch, and you'd been laying there too pale and too thin and dizzy with fever and not breathing right and... well, there's some stuff even he can't ignore.
So now here he is, trying to figure out how the heck to take care of you. He's got you bundled up on the couch in his 'office' area, because he doesn't want to let you out of his sight. But he's actually debating carting you up to your bedroom, even if it would mean he had to leave his work down here, because it's not like he's actually getting any plotting done right now, is it? He's got Johnny out on a supply run getting all the... things, one needs when somebody is sick. Cans of soup and bottles of cough medicine and... what the hell else, fever reducers and crap? Frost'll know. He can trust his man to handle that.
But supplies are one thing. He's not letting anybody else near you when you're like this, hell no. He's not letting anyone see him playing nursemaid, either. Because he's got a reputation, but with you? Sometimes he can be a little soft.
"I told ya, Pumpkin, you're gonna stay here and rest. The boys got things handled outside. You warm enough?"