Coeliacs or an allergy to gluten tends to be genetic in nature. My sister's oldest (whom I think I have mentioned before) was diagnosed when she was about two - neither my sister or her husband have it but were told one of them were a carrier of the gene and have passed it on.
Its one of those diseases thats pretty tricky to diagnose - the blood tests aren't very sensitive and usually only show up a positive result when there has been a high amount of gluten in the digestive system (hence it is a better indicator in adults than children). The gold standard for diagnosing is still doing a biopsy of the gut. It took ages for my sister to have it diagnosed for this very reason until FINALLY her daughter had a biopsy which confirmed it.
Unfortunately, cutting out gluten may 'fix' the problem, but Elle will need to have it in her diet if you want a definitive answer through testing (and also, having a child on a gluten-free diet who actually ISN'T coeliacs is just as harmful as having a coeliacs eat gluten - if that makes sense

)
I think you have to have perseverance when it comes to coeliacs - its difficult to diagnose and you may have to go to several doctors before you get anywhere. Some symtoms my niece had though was lack of growth (weight/height/muscle tone) over a year or so, bloated tummy, constant constipation or bouts of severe diarrhoea/vomiting and generally grizzly.
Hope I didn't confuse you...