Ok, first off it's just plain wierd to see an EMH in sick bay with an illness. I don't know how much work this would be, so maybe it's not possible, or maybe it can be a longer term item but here's how I'd handle an EMH injury.
When an EMH gets injured, that EMH becomes unavailable like now, but the next thing you notice is a shipboard engineering mission to repair the EMH. A failure would result in the EMH continuing to be "off line" and the mission again becoming available. A disaster would wipe out your EMH, while a very slim chance critical success would bump up the rarity of your EMH by a level.