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.
Personally, I would follow the instructions for wiring it as if it did have a light kit. You, or someone else, may someday want to add one, and that's far simpler if you can just remove the cap, plug in the connector, and bolt the light fixture in place. They may still want to rewire in order to put the lights under control of a wall switch rather than or in addition to the pull chain, but if not...
Best Answer
If the chain is broken inside the housing where you can't see it, you will have to disassemble the lower light cover on the fan if so equipped(usually 3 small screws). Definitely turn off the power to the fixture and always double check with a voltage tester to confirm power is off. It is not uncommon for the chain to break or actually pull out of the switch assembly. Until you determine the extent of the problem, you can't plan the fix. If the chain is broken at a ball link, a simple snap link my solve your problem. If it is pulled completely out of the pull switch, then you will have to remove the wire nuts and switch and get a replacement, usually available at any hardware store. Reassemble, carefully tighten the new wire nuts, a safety wrap of electrical tape around the wire nuts and put cover back on. It is actually a very simple fix if you have any experience with electrical repairs. CAVEAT. Be sure power is off before working with any wires. Good luck.