IMO Chemi-Pure will do neither in any measurable amount. It will not remove phospates or silicates in an amount that will make any difference. I am speaking about the Elite.. the regular Chemi-Pure doesn't even do Phos/Silicates, but the elite does, but just slightly.
To really remove phosphates and silicates, you should run a Phosphate reactor and use RowaPhos. I used Phosban for years thinking I was doing good, but I always seemed to have a silicate problem even though Phosban stated it removed them. Turns out, it doesn't.. well, again, not in any real measurable amount. This should be done after you have rid your tank of cyano first though.
So, if it were me, and I did go through this just a few months ago when I upgraded my lighting, I went in this order.
Step 1 - Run Chemi-Clean for 48 hours, turning off skimmer and remove carbon during entire treatment. Do a 50% water change, then repeat for a second dose. Takes 4 days total.
Step 2 - Run a bag of Phosguard for 3-7 days to remove any excess phosphates and silicates quickly. The media fills up fast so it doesn't last long.
Step 3 - Install a Phosphate reactor and use RowaPhos media. Also put Chemi-Pure back in at this time. I use 250g of Rowa media in my 29G tank. Make sure to rise extremely well and put a filter bag at the end of the output hose to catch any media that may escape so it doesn't get into your tank. The trick I learned over the years is put the reactor/pump in a 2 sided kitchen sink, plug one side, fill with water and place reactor/pump in that side, have the output hose going out the other side of the sink, run this for around 10 minutes to clean good. Make sure the flow is set right, you want the media to just be perculating in the reactor, not blowing around.
Just my 2cents on how to solve your problem, and I know it works because I had the same problem in Feb and solved it completely in under 2 weeks without any problems in the tank.