Here is the way the stimulus money works:
Each department gets awarded a certain amount of money, and they more or less get to decide where the money gets spent. However, any stimulus spending must then be approved by the OMB, a section of the executive branch. This is why it takes so long. Depending on how anal your rep in the OMB is, getting your desired expenditures confirmed can take a long time.
Also, the stimulus money only needs to be allocated after 18 months, not actually spent. So stimulus money can be used to pay us in 3 years because it was allocated to us this year.