For those two devices you could use a small 150 watt inverter, which is small enough to use a cigarette-lighter socket if you don't want to wire it directly to the battery. Anything bigger than 150 watts needs to be wired to the battery, via a suitable fuse.
You need to be aware of just how much power a heating device like a hairdryer or kettle takes. In contrast to the 65W total of your TV setup, a hairdryer takes 800W minimum, and can be higher. A microwave needs over 1000W. A standard domestic kettle is 2200W, and a low power one is 1000W.
There is a downside to larger high power inverters, in that they drain power even when nothing is plugged in. They need to be switched off when not actually being used.
I would get a small 150W Pure Sine Wave inverter to power the TV setup. A bigger inverter requires more thought, and possibly more/bigger batteries and/or solar.
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