The first phrase "I will be busy every day FOR the next two weeks" is better/correct since it indicates a period of time for which you will complete an action. Other examples: I have been on vacation for the past seven weeks. I'll practice my language skills for an hour each day.

You can also just say "I will be busy for the next two weeks." But if you want to stress that that includes each and every day, you can say "I will be busy every (or: each and every) day for (or: over) the next two weeks.