て adds the perfect aspect, which is more natural as suggested by the English version using perfect tense. 来週には忘れる is less natural, but not impossible.
Regarding は, you are right. This is because the sentence means 'by next week'. 来週忘れる would mean 'I will forget next week', which is unlikely (forgetting is not a conscious action; as an side, 来週に does not fit in this case).
Cf. Suppose you have a task that takes another week. Then, （この仕事は）来週には終わる sounds (to me) as natural as 来週には終わってる. Strictly speaking, the former suggests that the task does not finish this week while the latter does not specify - but I guess this is a technical differentiation and not that we really use them differently.
If you talk about a TV series finishing next week, （このドラマは）来週終わる is more natural than 来週には終わる mostly because the program is usually once a week.