It's not almost done, it's barely begun. They had a big scale model made, and maybe they've started digging up the foundations. Not much more than that. The Ogier refused to help, and Elaida had problems finding masons (and of course, no sister would have helped with the construction, Elaida didn't even think about that solution...).

At the technological level of the WOT world (without OP), finishing a building like Elaida's palace takes many, many years - often decades. For e.g., the Ogier haven't finished repairing the towers in Cairhien, 20 years after the Aiel war.

The building as planned would most likely be totally impractical to house the Asha'man. People tend to forget this isn't a second White Tower Elaida was building. It's a palace of only a few levels, built over what used to be the Warder's practice grounds. Elaida wanted a giant spire, higher than the Tower itself. Elaida certainly intended it to be more lavish than any national palace, but it wouldn't be a building to house several thousand people.

It's possible a building for men ends up being built in its place now, though the Tower itself was already built to house male and female channelers together (2 wings of everything). Obviously, the original builders had not yet given up all hope of cleansing saidin.