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Mycroft Holmes almost never gets ill, but when he does, he gets ill very thoroughly. Oh, he'll start out trying to make it through the day, but very quickly he'll learn he needs to cancel appointments and lock himself away.
He'll sit and read or watch telly, he'll roll over and try to be comfortable, and when he feels particularly nauseous, he'll dim the lights and play audio books lightly in the background.
He knows not to attempt work. All he'll attempt is calling in his physician for a check-up (house calls cost a bit, but it's worth it to not have to go anywhere).
He'll either start texting instead of calling or just call Sherlock anyway, just to keep tabs on him as usual. Either way, Sherlock will catch onto the fact he's sick, and that's when he'll come rushing over (if he doesn't have a case on).
Sherlock will sit with him, read to him (which he much prefers to the voices of those he doesn't know), will help him talk through what long-term strategies his head feels too clouded to keep on top of, will even help him to the bath.
He's a very dear little brother, at times. When they were younger, Mycroft helped Sherlock when he was sick, of course. But Sherlock was a hard patient, always pushing against the help, always sulking, always snapping. Mycroft isn't like that at all; if he was, Sherlock never would have helped him. But he always had, and he always does.
He'll even slave over the stove to make soup, which turns out well because he's had much practice. It didn't turn out well when he'd started, but then, he'd been very young.
The best thing about being in Sherlock's care is when he strokes Mycroft's hair and says, "You're such an idiot, getting sick like this." Mycroft will smile and let his eyes close, knowing it really means, "I'm here for you, brother, and I love you more than I'll say."