Women do indeed have souls according to Islam. The rumour that women have no souls in Islam is just that and you really shouldn't take rumours as facts.
The truth is that Muslim women were given more rights centuries before Western women got any. Hell, even black men were allowed to vote before women could in America. Women are treating so well in Islam (not necessarily in Muslim countries but we blame this not on Islam but on misogynist interpretations of Islam) that for every man that converts to the religion, four women convert.
So tell me, why would so many Western women willingly convert to this religion that, according to you and a lot of other such people who know nothing about Islam, hates them?
“I will not suffer the work of any worker among you to be lost whether male of female.” (Quran 3:195)