(Before I forget, typical solder melts at under 400°F, so you can't use regular solder to attach your coil or anything else that will be at that temperature. You can get heat-resistant solder, but that will probably require a special soldering iron.)You effectively have two parallel designs. Heating element and regulator. Each one is relatively straight forward.
Lets start with heating element. This is effectively just a NiCr coil, (I agree, that's a good choice) and a battery. As an example, let's look at battery you linked. It has two vital parameters.
Capacity: 850mAh
Discharge rate: 20C
20C tells you that this battery can spit out its charge in 1/20th of an hour. The capacity of 850mAh tells you that for 1 hour it can hold current of 850mA. So the maximum current it can withstand is 20 * 850mA = 17A.
That's a lot. At 3.7V, which is the output of a single LiPo cell, that's 17A 8 3.7V = 62.9W of power. Think mid-range incandescent light bulb. We are definitely going to go down from here, but it's good to establish an upper limit.
Wikipedia article on
NiCr has many useful tables. The current-temperature table is your first stop. You definitely want a higher maximum temperature, so that the heater can kick in fast enough, but you probably don't want to go past 600°F. Note that for none of the gauges that puts you over 17A, which is great. It means that you aren't going to overtax the battery.
I would also be a bit worried about overheating your wires. You want to keep the current through these relatively low. The 600°F at 22 gauge wire suggests 3.63A of current. That seems reasonable. It would require resistance of 3.7V / 3.63A which is just over 1 Ohm. So now you want to determine wire length that will give you 1 Ohm.
The resistivity is listed in properties in the first table on the page. It's given at room temperature, but the resistance only increases by about 5% in your range, depending on composition, so it's close enough. To figure out resistance per length, you need to know cross-section area of your wire.
This table should help. 22 gauge is 0.33mm². And resistivity is given as 1 to 1.5x10
-6Ohm*m. That's quite a spread, so you'll need to look up what it is for specific wire you'll be using.
Lets go with 1x10
-6 for simplicity. Resistance per length is resistivity divided by area. Keeping in mind that 1m = 1,000mm, 1m² = 1,000,000mm². So 0.33mm² = 0.33x10
-6m². Dividing resistivity by area, you get 3Ohm / meter. So for 1 Ohm of resistance, you need 1/3m, which is just over a foot of wire.
These figures will change depending on specific wire you go with. So look up these numbers, follow these computations, and you should be able to derive the wire length for your particular project. Then coil it up, making sure it doesn't touch itself or any other metal anywhere along, and you should have your heating element.
Ended up with length that's too long? 1 foot is probably excessive for your application, resulting in rather large coil. What you need to do is go to thinner wire. Instead of 22 gauge, maybe try all these computations for 24 gauge. That will both reduce current needed to heat up the wire and increase the resistance, resulting in shorter wire needed and a smaller coil.
Ok, so that's pretty much what I can tell you about the heating element itself. Let me get some thoughts together and I'll write up something on thermal regulator.
