Kindle uses the Mobi format. Calibre's mobi conversion should work on the kindle
Unable to reupload any books due to personal reasons
^^ As above, it shouldn't be a problem converting them all to epub using calibre
I download predominantly epub books, because I sometimes read on my phone as well, and epub works better with Moon+ Reader in my experience. No problem with conversion via Calibre. Calibre also has the option to auto-convert epub into your preferred format once you add the books into your calibre library, which is convenient 
Kindle Paperwhite supports KF8, which is basically an ePub wrapped up with a .mobi file for older Kindles.
So if you install the Kindle Previewer, you can set it to Paperwhite and import your ePub into that. This should then convert it to a KF8 .mobi file (I've only used the Kindle HD settings, so can't be 100% sure - I'm still having a play with it as I'm looking into adding an .azw into future releases) which it will put in a new folder in the one that the ePub was in. It'd be worth trying even if you had an older Kindle, because, as jaysee points out above, there are slight formatting issues and this will fix that too. Of course, if all you want is a readable ebook and you are already using Calibre (as the majority of we ebookworms do) then Calibre is the best for your requirements.
edit: Looking it up and I see Calibre has an AZW3 option in its conversion options, so it might be worth looking at that, although I am unsure if this will fix the .mobi formatting issues. At least with Kindle Previewer you'd know that you are getting the full official Amazon-approved conversion, even if differences might be minor. I'll have a play with this too and see what happens.
So if you install the Kindle Previewer, you can set it to Paperwhite and import your ePub into that. This should then convert it to a KF8 .mobi file (I've only used the Kindle HD settings, so can't be 100% sure - I'm still having a play with it as I'm looking into adding an .azw into future releases) which it will put in a new folder in the one that the ePub was in. It'd be worth trying even if you had an older Kindle, because, as jaysee points out above, there are slight formatting issues and this will fix that too. Of course, if all you want is a readable ebook and you are already using Calibre (as the majority of we ebookworms do) then Calibre is the best for your requirements.
edit: Looking it up and I see Calibre has an AZW3 option in its conversion options, so it might be worth looking at that, although I am unsure if this will fix the .mobi formatting issues. At least with Kindle Previewer you'd know that you are getting the full official Amazon-approved conversion, even if differences might be minor. I'll have a play with this too and see what happens.
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Strange is the night where black stars rise,
And strange moons circle through the skies,
But stranger still is Lost Carcosa.
Strange is the night where black stars rise,
And strange moons circle through the skies,
But stranger still is Lost Carcosa.