I badly injured my heel a week ago when I jumped down cement stairs from way too high. An X-ray showed it not broken, but it is still very painful. How long will it take to heal? When should I see a doctor again and what kind?

It take a while to. Heal, elevation, motrin, and warm moist heat will help. See an orthopedic surgeon.
You Need to. See a podiatrist or orthopedist and I would commend either a ct scan or MRI to rule out injury not revealed on x-ray. Or I should say possible injury......
Heel fractured? It may be worth repeating an x-ray or obtaining a different type if imaging (ct scan or mri) to see if you have a fracture that did not show up on the original x-ray.
See a specialist. You need to see a foot and ankle specialist, podiatrist or orthopedic foot and ankle. Sometimes heel fractures can be subtle and easily missed on x-ray- you may need a ct scan. Till then rest, ice, and elevate. Also, stay off of the injured foot and see somebody soon.