All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
If these lost items are simple and people do not usually care about them, then there is nothing wrong for whoever finds them to take them and make use of them without making them known (to find their owners).
However, if it is something valuable which the average person usually cares about, then this is a lost and found property that must be made known for a whole year.
After the passing of one whole year, it becomes permissible for whoever found it to benefit from it; please refer to fatwa 133112.
If the gym management throws these items in the garbage after one month, this is not permissible, but you are not allowed to take them unless you want to make them known.
Allah knows best.