Determining the chemicals in potash and crystallizing them
I think it would be quite difficult to separate all of the different compounds dissolved in the sample into their solids because they are all in the same phase. I do not think any method exists for this kind of separation. 

However, determination is absolutely possible. 

Firstly, regarding your deductions...

Although carbonates give carbon dioxide gas, which is colourless and odourless, it is nevertheless important to test that the gas is indeed carbon dioxide by bubbling it into limewater and looking out for white precipitate. If you have not conducted that test, then you cannot claim that the gas produced is carbon dioxide. Otherwise, you can conclude that there is carbonate in the sample. 

The high pH is indeed indicative of high hydroxide ion concentration in the sample. That is for certain. 

However, you have not proved that the cation is indeed potassium. Perhaps, you want to conduct a flame test or do analysis using atomic absorption spectroscopy (AAS). Procedure for flame test can be found here: https://www.soinc.org/sites/default/files/uploaded_files/flametest.pdf

If you would like to find out what other ions are present in the sample, carry out a simple qualitative analysis using some common lab reagents. Procedure can be found here: http://www.wiredchemist.com/chemistry/instructional/laboratory-tutorials/qualitative-analysis

By boiling off the water, you would likely get some ionic solids but I believe that it should contain a lot of impurities.