I've been helping a relative handle this and these are our solutions:
1 In the short term, show the school the receipt part of prescriptions for inhalers and other medications - that's the part the pharmacy puts in the bag with the dispensed prescription;
2 Keep all receipts of any over-the-counter medicines purchased, eg Calpol, calamine lotion;
3 Make sure you're signed up as a family to every online facility your GP surgery offers: summary records, medication ordering, e-consults, emails, everything.
You can print and keep everything and create a file.
It was this approach that meant we were successful in obtaining (low rate) DLA for a very young child in our family with severe asthma and allergies.
You can also then revert back to the school and ask them, in terms of your DD's asthma, what the school's policies are re risk assessments and ECHPs. My relative's DD has extremely bad asthma attacks and everyone needs to know what they're doing, quickly.