Solid premise with a couple of neat ideas that could really shine with a bit more interactive detail. Understandably towers are tricky to do because they tend to be very linear.
I felt the first room was an unused opportunity as it’s completely barren besides the secret door, which is so strongly telegraphed by the lone chair, it’s not really a secret at all IMO. This would have been a great spot for an interactive element, like possibly a basic puzzle to locate the door (choose/sit on the right chair, with the incorrect choice having consequence); provide an additional clue like a faint hand print or bloodstain, etc. Something that would make the PCs interact and investigate rather than take the initial scene entirely at face value. Even adding a simple detail like a tapestry or mural about the archmage help make an empty room much more interesting. :)