This is not a strong approach.

1. The major mistake is the following: you assume an average of children per woman, which is the same as assuming that EVERY woman will have 2 kids. 

In real life, you have a good level of varuation. Zero kids, one kid, three kids, etc. Making an obvious exageration, you can have an average of 2 with 50% of the woman having 4 kids.

Therefore you should be considering woman with no kids, with 1 kid, or with 2+ kids.

2. I also don't like much the idea of dividing population by average life expectancy. At least not without stating a few relevant assumptions, as this assumes stable population for a long time, and this is only true if you assume that fertility rate has been close to 2.1 for a long time, or that any value below has been compensated by immigration). 

This is critical because if you afterwards assume a value different than 2 or 2.1 for fertility rate, then you will have an internal inconsistency. 

On a minor note, this also assumes that every person lives up to the stated life expectancy (this is the same mistake as on number one). While this is not a huge mistake in the developed world, it is significant enough that the recommendaed fertility rate to maintain population is 2.1 and not 2.0. So a 5% difference. Not a huge mistake, so I wouldn't be too concerned about this one.

Hope this helps in steering you towards the right direction.