Time--hard to say. It depends on several things, but probably a while. Washing soda is freely soluble in water and dissolves fairly rapidly with stirring. Ca(OH)2 is not very soluble in water, and the dissolving of it will be slower. However, as the washing soda dissolve it should react with the lime to produce product and more lime will dissolve. It also depends on of the physical shape of the Ca(OH)2. If it is a chunk it will take a while for the reaction, but if it is a fine powder, the reaction with washing soda should be fairly rapid. Also, there is no mention of temperature. If the water is very hot, this definitely speed up the reaction. So the time question cannot be easily answered without more information. NaOH is called lye and is used in making lye-soap, and it is fairly inexpensive and easy to obtain. It would be much easier to just buy some lye. To make it in the way you are asking, will probably eventually work, but that is an awful lot of trouble when you may be able to buy lye at a drug store or hobby shop or soap making distributor.
You method would totally work. We use that reaction to knock the calcium out of water in the cold-lime softeners here where I work, in up to 4.5MGD. We do add iron and flocculants to help the caco3 fall out of solution quicker/make it easier to remove. Evaporating the water would also take a lot of time/energy. I totally agree with the other answer, just go buy some if you want it. Powder/solid drano, which you can get at any grocery store, is basically pure lye.
