Electric brewing sounds like the best method, but with quick research it seems to have high up front costs. Nearly as much as I'm spending on basic BIAB equipment.
Has anyone setup electric brewing for $150 or less? Is that impossible?
Well it all depends on your wants and needs. Most electric set ups referenced here are more sophisticated and of course cost more money.
You could install a 2000 w element in your 11 gallon kettle and make beer very inexpensively. Yes it would be slow to boil, and take time. It would afford you the convenience of indoor brewing.
The hot rod heat stick mentioned earlier coupled with a kitchen stove is also a good option.
I find carting equipment outside to brew a royal pia, and would much prefer to brew bare bones electric inside than setup propane outside.
Two 1500w elements in your 11 gallon kettle will boil a 5 gallon batch nicely ime.
You would need two separate 120v circuits w gfci.
240v elements running on 120 produce 1/4 of the voltage, so another option is to run a 5500w 240 at 120 to produce 1375w times two would be 2700w.
My hunch is you will need b/w 2500-3000w total.
Doing this would be well below $150.