I can understand why she was upset as she is now in a hotel with a sick child and none of her usual resources (medicine, hydrating drink, etc). However you had no other choice really.
What is surprising to me in the story you gave is that she seemed to have multiple indicators that she should not have come and she ignored them all. Her child had not been drinking for a couple of days, was cranky, was showing signs of feverishness. She shouldn't even have left home, if only for the sake of not making her unwell child endure a long train journey and the risk of being outside her normal space during an illness.
But then, once she arrived at your house, she downplayed multiple signals that her child was unwell and that this was a communicable virus (fever, vomiting) and seemed to think that just giving Calpol would somehow prevent your child getting the virus? The vomiting alone probably sprayed virus particles everywhere. So, I feel badly for her but she made a big error of judgement, considering you had made it very clear from the beginning that you can't have sick people in your house.
I think it was fine that you paid for the hotel, (even though her error was the cause), because emotionally it must be very unpleasant for her to be in a hotel with a sick child. But I would not apologise for asking her to leave as you did give her fore-warning of your needs and she just ignored the warning signs and did exactly what you asked her not to do. Things would have been a lot simpler if she had simply called you this morning before she left her house, and said, "Kara is off her food for a day or two and seems a little cranky. She may be coming down with something, or just having a bad day. Should we risk it?" And you could have said, "I'm so sorry but I can't risk it, let's reschedule."