You are most welcome to HCM,
I have gone through your question.Elevated LDH is not specific and may occur in many diseases.
since January, on & off without any definite diagnosis, might be PUO. Now, Straightly answering to your question i.e., "What else should we test for?", I personally believe that you should go for (for your 12 year old daughter) -
(1) Rule out TB (Sputum for AFB, CXR-PA/Lat, ESR
(2) FNAC thyroid
Consult it with an specialist before proceeding.
Thanks and I wish your child recovers soon. God bless her.