Your tendon isn't just inflamed. It's under-supplied.
Long-lasting tennis elbow usually isn't a simple case of inflammation. Researchers describe it as tendinopathy: the tendon's fibers slowly wear down and struggle to repair.
Part of the problem is supply. Tendons get far less blood flow than muscle, so oxygen and nutrients reach them slowly. That's why ice and anti-inflammatories often feel good for a day, then change nothing.