I don't think the replies here are quite fair.
Legally, yes, she can state she has an illness and there is nothing you can do. As Op stated, her hands are tied.
But yes, there are people who will use that for their own advantage, and it's not fair that the honest people effectively miss out. I think most people know people like this.
I used to work with someone who if any thing came up that she didn't want to do or someone actually had the courage to point out that she wasn't pulling her weight (which she didn't), then she'd say straight that if it wasn't changed/they withdrew what they'd said, she would tell the doctor to sign her off with stress. She was totally open about it and would tell people she'd done it and why.
On one occasion she was asked to join a project and didn't want to do it (I suspect because the manager of the project was known to be not afraid to say if he felt someone was underworking) and she immediately sent a message saying that she was off with stress and wouldn't be back until end of April (over 2 months later), which was when the project was due to finish. Her doctor would only give sick notes for two weeks, so she had to keep going back to get another one. On the day after the project was due to finish she arrived in work to "see how everyone was" oka check the project had finished.
Another time she announced she was having to go off on stress until mid September - she was about to go on 2 weeks annual leave, and then that was another 2 weeks after that. Turned out she had a holiday booked the 2nd week in September (booked beginning of August, the day before she announced she was going to be off with stress - we know because she was going with a group from a hobby, which happened to include one person's sil) and she'd run out of annual leave.
And despite making it as obvious as she could that it was entirely made up, there is still nothing HR could do as long as the doctor signed the notes. She even told one of the managers in her "return to work" meeting that she was fine and had just had "things she needed to do at home".
What happened ultimately was the rest of the workforce have mostly moved on, with a large number saying in a good part due to her behaviour (not just on this, but other things too), but she is still there doing very little work, and no one daring to pull her up because they know what will happen if they challenge her.
And people like that annoy me. Because not only do they get away with anything, leaving other people taking the slack. But it means that people who really do have medical issues that need support, people find it harder to believe them, or not resent them.