I've had graves' disease for a year and a half now, how long does it take for someone to be cured from graves' disease?

Never: Grave's disease is an autoimmune condition which cannot be cured. However the thyroid function can usually be regulated.

It depends.: If you are taking anti-thyroid medications (commonly ptu (propylthiouracil) or methimazole), the goal is usually control of the graves', not a cure. To become cured, you would need to consider removal/ablation of the thyroid, either with surgery or radioactive iodine.