The white TV power button on the left of the remote allows you to turn the TV on/off independently from the 360 set top box. Press once to turn off the TV and twice to turn it on.
The red set top box power button on the right of the 360 remote allows you to switch the 360 on and off.
If One Power is enabled (Settings > Audio & Video > One Power) you can turn on most TV's and connected devices at the same time as the 360 with the red power button, but only if they are compatible and connected using an HDMI lead.
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I am very sorry to hear you're having some issues with your TV remote recently. The white power button on the top left will be to turn on the TV, and the red power button in the top right will be for the TV box.