If i'm taking medications properly, how long will it take for hashimoto's thyroiditis to go away?

It depends: It depends upon what you mean by "hashimoto's thyroiditis". Hypothyroid symptoms should clear up within 6-8 weeks of treatment with a dose that makes the TSH normal. Inflammatory symptoms (pain & swelling in the thyroid), get better in in most cases within a few months, but can persist for years in other cases.

Hashimoto: Highly variable but response to treatment should definitely become apparent in 4-6 weeks.