Here is a good diagram that shows how a 3-speed switch works.
To control the fan speed without the switch, you'd have to apply current to the proper wires to get the speed you want. Since this is likely a mechanical device (not electronic), the only way to change the setting is to physically move the switch to another position.
A tiny bit of chain is visible
If even a small bit of the chain is left, you may be able to pull it using needle nose pliers. If the fan is connected directly to power (not controlled by a wall switch), you'll probably want to shut off the breaker to stop the fan while you do this (so you don't loose your head). Most fans have the same click pattern.
- 1st Click = High.
- 2nd Click = Med.
- 3rd Click = Low.
- 4th Click = Off.
So be careful to count your pulls, or you'll be making another trip to the breaker.
The chain is just GONE!
If the chain is completely gone, the easiest option would be to replace the entire fan.
If you like to tinker; and have a bit of knowledge about this stuff, you could take the fan apart and replace the switch. It's probably a good idea to have the old switch in-hand, when you make your trip to the hardware store for a replacement. Don't forget to label the wires when you remove the broken switch, you won't remember where they all went.
Note: Flipping the breaker on/off a couple times, will do nothing but turn the fan on/off a couple times. It is possible to control the speed of the fan by continuously flipping the breaker on/off, but your arm would get tired pretty quick.