Right kind of resistance wire for near-skin heater?
Your desired temperature is low, but the wire is small so will get significantly hotter than the overall temperature increase.  Nichrome is probably a good choice.

However, the real problem is the total energy required and the size battery that will take.  Maintaining let's say 10&deg;C additional temperature rise over the surface area of a jacket will take some power.  You'll have to measure that.  My guess is it would take 10s of W.

Another problem will be the temperature of the wire.  You'll probably wrap the wire in a zig-zag with the spacing being much larger than the wire's diameter.  This means the wire itself will be significantly hotter than the average temperature rise.  That could be a problem if it directly touched the skin, or it could melt, otherwise damage, or even catch the insulation on fire.

Yet another issue is that you don't want the inside of the jacket to be 40&deg;C.  That's above body temperature.  That will get uncomfortable fast and cause sweating, which is really bad in a winter situation.

Go get a better jacket instead.