How to power an LED with a thermocouple
There are a number of IC products made for this purpose, called &quot;energy harvesting&quot;.
For example, Analog has several, one of which is shown below:

A single type J junction (iron - Constantan) requires a temperature difference of around 500-600°C to get 30mV (typical start-up), that particular chip requires 50mV to be guaranteed start-up.
With 10 type J junctions in series you can guarantee start-up with a temperature difference of only about 100°C  (the more current drawn, the higher temperature difference required depending on the thermocouple resistance). Type K (Chromel-Alumel) is not quite as high output, but it is less prone to corrosion .
To get lower (electrical) resistance requires thicker wires, which means more heat flow between the hot and cold junctions. More junctions (electrically) in series means heat flow is multiplied proportionally since they are thermally in parallel. As always in thermodynamics, there is no free lunch  and you need to keep the &quot;cold&quot; junction actually cold enough relative to the hot junction with a suitable heatsink or maybe even a fan. Overall conversion efficiency will be rather low compared to Carnot.