Testing for asbestos using readily available chemicals and tools?
Is there a chemical test for $\ce{[Si2O5]^2−}$ that can be done at home?


First $\ce{[Si2O5]^2−}$ isn't a poly atomic ion like sulfate ($\ce{SO4-}$) or nitrate ($\ce{NO3}$) but represents a stoichiometry in the material.

Unfortunately at this time a test without x-ray diffraction is not possible. Asbestos/chrystolite is not just a chemical formula, but a crystal structure. It's not just what it is made of but also how the atoms are structured that makes it potently carcinogenic. If you were to take asbestos and melt it then cool it sufficiently fast to form a glass (amorphous/non-crystalline substance), it would not have similar carcinogenic properties as the crystalline material.


  I am aware that one test involves viewing the sample under a sufficiently high-powered polarizing light microscope, but not everyone has such a tool.


It is worth noting that using polarized can be done at home with polarized filters but is a presumtive test not a direct method. Essentially all chrystolite will rotate light but not all minerals that rotate light are chrystolite.