My older son (now 6) sounds similar, needing to be busy, and also wanting interaction a lot. He's in full-time K now, but preschool was just a few mornings a week. Our setup isn't perfect but... We have a few different areas in the house where he can find a different activity:
-We have two kids' desks (we also have ds2 age 4) with drawing/workbooks/coloring supplies, one is in the dining room, one in my "office".
-There's a book nook in the living room with a bunch of our books and about 40 library books at a time.
-There's another table where they can do "activities" that kind of rotate and get put away when they seem bored with them - Snap Circuits, weaving loom, play dough, etc.
-There's a CD player with kids' CDs that they have access to.
-Recent addition is a digital piano (thank you, craigslist!), but there's always been a box of little instruments - xylophone, harmonica, shakers.
-Low shelf in the livingroom of puzzles/games in boxes. There's not much in the bedroom upstairs but there are a few books and puzzle-y toys.
And when he's seeming crazy (maybe bored I guess?) I tell him it's outside time!! And I send him out. He goes on the swings, or scooter, or recently hula hoop sometimes. Even just 10 minutes refreshes him. So, basically there's a lot to do in the house, but I help him along with ideas and sometimes insist he go outside, which helps a lot. At 4 I started letting him go out by himself as long as I could see him in the backyard.
Hope this helps at all. Consider rotating out the stuff in your drawers for her?