OP, consider a different setup to show what's in your garage on your television. You're asking a lot of HDMI to pipe it that far. And that camera is a camcorder that is probably rarely used as you're trying to use it.
If this is some kind of one off thing- like you're trying to shoot some kind of mechanics video and want to use your TV as a monitor- consider moving the TV closer to the garage until you complete the shoot. Or shoot the scenes in camcorder mode and then take the camera to the TV to proof them.
If this is some kind of ongoing thing- where you want to regularly monitor the garage for security or similar reasons- there are many other home security options that will pipe video via wifi or ethernet over upwards of great distances. HDMI is not really intended for long runs. Consider seeking alternatives that are made to do what you are trying to do. Go to Amazon and search for "HD Wireless Security Camera" for a plethora of cheap and no-so-cheap options.
If you just must get that setup working, eliminate the possibilities by testing each part. For example, if you have something with a regular HDMI port on it (like a BD player or a laptop), try the long cable by itself and see if the picture reaches the TV. If so, that will point to the mini hdmi adapter (but only in how it works with the long cable since you already know it will work if attaching the camera directly to the TV).
Good luck!
