What a nightmare for you! I hope you have made a full recovery.
I think a panel will fully understand that your daughter will not have been at her best for the first paper, and as this was the out-of-county test I am sure you felt very alone in trying to decide what to do for the best.
Can I just check that the scores were in the order 111/116? Also, how did your daughter feel after each paper, and were they taken on Monday and Wednesday? Had you begun to recover by the second test?
Her SATs scores seem very encouraging, so please now read the Appeals Q&A http://www.elevenplusexams.co.uk/appeals
thoroughly if you have not already done so, talk to the Head, gather together as much academic evidence as you can and then come back to us.
When you see the Head, please check if s/he completed a Head Teacher's recommendation form in advance of the tests. Not all out-of-county Heads do so.