Mitochondrial? In many instances, EDS hyperextensible type is caused by an underlying Mitochondrial condition. The mitochondria produce all the energy we need for and in every cell of our bodies-Progressive daily fatigue, muscle/joint pain, dysautonomia,autoimmune disease. http://www.umdf.org/site/c.8qKOJ0MvF7LUG/b.7929671/k.BDF0/Home.htm.
EDS. I assume it's ehlers danlos you're reporting. I don't know of any association w/ hypothyroidism. Proceed with treatment. It is straightforward with the usual course being added T4 in a pill. Monitor for symptoms of thyroid slowing, some if which overlap with eds. Soft tissue swelling, joint problems. , there may be others.
Continue evaluation. Pituitary problem can cause both conditions so i would talk to your doctor to make sure this is not the case (blood tests can figure this out). If no problem with pituitary, you can treat both conditions at the same time or treat the hypothyroidism first and make sure it is not causing the eds. If the hypothyroidism is fixed and eds is still there, you can treat it then.