It's all very well to say have back up childcare...but from my experiences - and I say this as a single parent of 2DC (eldest being 18 now!) - most childcare settings will not take a child who is poorly...they certainly didn't when my DC were younger.
My DC went to nursery, CM, and once they started school, used the school wraparound care and holiday clubs. I also used the Sitters website to cover weekend shift work. I had childcare sorted for most eventualities (plus back up stuff), but none of the places/settings I used would take the children if they weren't well (maybe things are different now?). A live-out nanny wasn't an option due to cost. A live in nanny/au pair also wasn't an option due to me having no spare room for them (and also cost!). If I had the money/a spare room in the house, a live-in nanny would have been ideal due to my shifts. FWIW, I'm a B6 nurse on a very acute ward- and I am not able to WFH. I do wish this was an option to cover management stuff (like audits, incident reports, appraisals, Teams meetings etc), but it isn't possible in my Trust (not for the likes of me anyway!).
My friends (who live locally) all work FT and have their own families - although, friends/colleagues and I do take turns to look after each other's DC if our shifts are different (but you can't always rely on that). My family is mostly overseas. My parents are both dead. My XH is/was not fit to look after the DC unsupervised (he's an alcoholic). My neighbours are retired but have underlying health issues. Who should then look after my DC when they're ill? My youngest is 11, and now can be left home alone for a few hours in the day (I do let neighbours know and she knows to call them/me if any problems), but I wouldn't leave her home alone if she was ill. My eldest would watch my youngest from when he was about age 14/15 (but again, I wouldn't leave him to watch her if she was ill). My DC are NT, have no underlying health issues and are responsible kids....so I have it bloody easy compared to many. But even then, it isn't easy if there is no one to watch DC if they're sick. When I help out friends with childcare, I'm flexible, but if the kid has D+V, I can't do it as I risk catching it myself/passing it to my own DC and then I/they would need time off school/work.
Before I had DC, I didn't understand/appreciate the issues that some people often had with finding childcare. Funnily enough, neither did two of my managers (who were bloody nasty to me...until they fell pregnant and asked me about the childcare available in our area for people who do work shifts etc!).
OP, aside from staying home to look after your poorly DD, I don't know what other options would be available, if you have exhausted the other available avenues. Also, I know the childcare options vary massively depending on the area you live in. In 18 years of being a parent - and despite living in an area that has four hospitals in the local vicinity - I found just one setting (CM) that even covered the hours I work on a long day shift (LD shift is 7am to 8pm). And at £220 a day, it wasn't cheap. And aside from my parents (who were retired), there really wasn't anyone who would take my DC if they were sick.
How is your DD this morning? I hope she's feeling better!