I am a bit confused on what you actually expected to happen after you changed your home page to https? Your opening post says your ranking stayed the same.
With regards to the technical side, you have few issues and it should be better that they are fixed:
- Internally you should link to https version of the home page.
- When you are on https version of the page, all other internal links should point to http version of other pages (since only home page is on https)
- If you have a sitemap.xml, the sitemap should only have https version of the home page listed
You say you have canonical link element that specifies non-www version as being your canonical. Is the canonical link element used on your https version of the page? I.e. if you open https version of the home page and do View Source, do you see any canonical and if so what?
What I gather the canonical link element should be is to point to https non-www version of the site (e.g. to [example.com
I would also like to add - the title of the thread is "Problem with Google reading HTTPS Homepage". Do you know whether Google in fact has the problem? From your posts (ranking unchanged for 2 months) I would say Google reads it just fine. Have you checked your logs to see if the https home page is being crawled? If you do a search on Google that would normally bring your home page in SERPs, what URL do you see in Google SERPs, http or https version?