Nichrome wire gauge calculation
The first thing you need to figure out is how much power per unit length it takes to maintain the wire at 43 &deg;C (110 &deg;F).  That depends on things you haven't told us.  The ambient temperature and thermal conductivity of whatever the wire is touching will make a large difference.

Calculating the power required to maintain a particular wire temperature is probably not possible since too many things will be unknown.  The best method to find the answer is to try it.

Get some wire, connect it to a lab supply, and see how much current it takes to keep it at the desired temperature.  This doesn't need to be nichrome.  Just about any wire can handle 43 &deg;C.  When find the setting that yields the desired result, record the current and the voltage drop across a known length of wire.  The current times the voltage is the power being put into that wire.

From the power into a known length, you find the total power into your desired length.  The power into a resistance is:

&nbsp; &nbsp; W = V2 / &Omega;

where W is the power in watts, V the voltage across the resistance, and &Omega; the resistance in Ohms.  If you are going to apply a fixed voltage, like your 3.7 V battery, then flip this around to find the resistance:

&nbsp; &nbsp; &Omega; = V2 / W

Now you look at wire resistance tables to find some that has the desired resistance over that length.