"Time management tools were developed by him for (the) staff"
”Time management tools were developed by him for staff” and ”The management tools were developed for staff by him” are both grammatically correct sentences.
(I’m of course assuming that you wanted to type ”developed” and not ”developee”
The vital thing to remember about English sentences is that the order of the words goes from most important to least important.
Firstly, note that both your sentences are passive forms. So obviously your question is about the ordering of words/phrases in passive form sentences.
Look at the active form of both your sentences:
”He developed management tools for staff”.
The order of the elements gives the order of importance. Here, ”He” is the most important, followed by ”developed” and then ”management tools”. ”for staff” is of lesser importance to the person communicating that sentence.
Now consider your first sentence:
”Time management tools were developed by him for staff”
So, the reason why you use the passive form in either of your sentences - instead of the active form - is that the primary object ”time management tools” is the thing of most importance you are trying to communicate. There are even uneducated native speakers who can’t understand this simple concept and always try to use active sentence forms!
So, ”time management tools” is the most important part being communicated and ”were developed” is the second most important element. ”by him” (i.e. ”he” transformed into the passive form) comes next in importance and ”for staff” is the least important.
Now, if you want to emphasise that ”for staff” is more important than ”he”, then you use your second sentence:
”Time management tools were developed for staff by him”.
In the passive form, you mention ”by him” because you still want to acknowledge that it was ”he” who developed the ”time management tools”. If ”he” becomes irrelevant/unimportant, then the form is:
Time management tools were developed for staff.
If ”for staff” is the most important part of the communication, then you might write a sentence like this:
”For staff, time management tools were developed by him”.
There are many other ways you could order this sentence to change the order of emphasis.