Another option is Reddit. There are sub -reddits dedicated to Batman fanfiction. You can post a request for 'batman sick fanfiction' there, and some fellow fans might be able to point you in the right direction or even share their own creations.
One could be 'The Dark Knight's Trial' where Batman has to face a new and extremely dangerous villain while being sick. It shows his determination even in a weakened state.
In many Batman fanfictions where Robin gets sick, it often shows a different side of the dynamic duo. It could be a chance for Batman to show his more caring side. Maybe he'd take Robin to the Batcave's makeshift infirmary and look after him with great attention.
One common theme is the bond between Batman and Robin. When Robin is sick, it tests and strengthens their relationship. Batman's protective nature over Robin really shines through.
Damian probably reacts with frustration. He's used to being strong and capable, so being sick makes him feel vulnerable. He might try to push through it at first, not wanting to admit that he's unwell.
One possible plot could be that Batman gets a mysterious illness that not even the Batcave's high - tech medical equipment can diagnose. As he gets weaker, Gotham's criminals start to become bolder. The Bat - family has to work overtime to keep the city in check while also trying to find a cure for Batman.
Maybe the other Justice League members will have to step up and take on more responsibilities while Batman is sick. They might struggle without his usual strategic mind.
Fanfiction.net is also a good place to look. There are many different categories and tags. Search for 'Justice League' and then narrow it down by adding 'Batman sick' to the search terms. However, the quality of the stories can vary, so you might need to read a few to find the ones you like.