I ran the electrical department in chain like The Home-Depot for 7 years. About once a month, someone would come in asking for a mercury vapor light set-up that would fit in a small area.
As for your situation: can you get into the attic above where the light is? If so, the proper way to do it is this:
get 5' of Romex 14/2 wire, 6 yellow wirenuts, two hammer-in Romex staples, two Romex-to-box connectors for a 1/2" knockout, and enough 3" or 4" flexible exhaust hose to reach the exit point (the attic louver vent - presumably on an exterior wall)
TURN OFF THE BREAKER TO THE CLOSET CEILING LIGHT AND MAKE SURE THE LIGHT SWITCH IS OFF, JUST TO BE EXTRA SAFE.
Bring a plug-in lamp into the closet so you have plenty of light to work with in there.
Remove the closet light fixture from the ceiling (usually two screws that hold it to the ceiling box. Twist the cone-shaped wire-nuts counter-clockwise to remove. Make note of the colors of wire that come out of the ceiling - black should go to black and white to white. There also might be a bare or green wire attached to a green wire or screw on the fixture.
IN THE ATTIC Mount the fan in the ceiling 1 foot away from the ceiling fixture, per the instructions that came with the fan. (you should have already cut the correct-sized hole from below using a 5/8 paddle drill bit and keyhole saw - making sure you are between studs)
insert the Romex connector in the 1/2" Knockout in the metal wiring box on the fan housing, push the Romex 8" into the box and tighten the clamp (you will perform the wiring connection from inside the closet)
staple the Romex to a ceiling joist within 12" of where it exits the fan.
Attach the flexible exhaust hose to the fan and run it to the exit vent. If you can attach it to the louvers with a sheet metal screw or wire or plumber's tape, even better (the REAL best way to do this is with a roof cap but you already said you don't want to do that, so I won't go into it)
knock out a 1/2" plug on the junction box the closet light is mounted in (if it is a plastic box you can just pull the Romex through one of the tabbed holes in the box)
push the Romex 8" into the closet light box.
Staple the Romex to a stud within 12" of where it enters the closet light box.
You are now done above the ceiling. GTFO of the attic. Those places are uncomfortable as hell.
In the closet, at the exhaust fan, strip the outside sheath off the Romex for 7" (be careful not to slice the insulation on the wires within. I usually make a 1/2" cut lengthwise from the tip, then use a fingertips or a pliers to peel it back like a banana, then cut off the dangling sheath with scissors.) Strip the ends of the wires and use the orange or yellow wire nuts to connect the Romex wires to the exhaust fan wires per the instructions. Place the vent grill on the fan.
At the closet light:
Strip the outside sheath off the Romex for 7" Strip the ends of the wires and lay the ends parallel with the matching colors of the closet light and ceiling wires. Should be three whites in one wire nut, three blacks in the other, and the bare wire from the Romex goes to the bare wire coming out of the ceiling or the green screw on the light fixture.
Mount the light fixture to the ceiling.
Turn the power back on.
Turn on the light and the fan should go on too.
ALTERNATIVE OPTION:
Go to Burlington Coat Factory. Buy her a big coat.
--TinyE