Thanks for the comments.

For the moment, I have thought about these things:

It could be useful to change the silica gel bags in the box after maybe
a couple of days. Then put the removed bags in a warm place so that
they would release some humidity from them. Then swap the bags again.
And try a longer desiccation time, as Corey suggests.

I also thought that I could remove only one half of the cassette shell and
pile the silica bags on top of the visible reels. Sometimes the tape 
reels are
quite loose, although it isn't too difficult to move them as they are so 
To make handling easier you can spool the tape somewhere in the middle.

I have had about 15-20 small silica bags in the box with the tape. The 
bags are
about 3 x 4 cm in size. I also have some a little bit larger bags.

I have kept the box closed, it isn't necessary that humidity from the room
gets into the box.