There isn't a 'great' solution here.
As I said, this is completely the 'wrong' cable and it was clear whoever installed it didn't know what they were doing. When I bought my first digital projector, about 20 or more years ago, I was already considering how I would deal with digital content in the future. It was not a concern at that time, but one thing I never did with that 20 year old projector? Hook up VGA to it.
Like I said, you can run a cable across the floor for a bit. You can ALSO consider converting from HDMI to VGA. This is less than ideal as you are making a digital to analog conversion, and the highest resolution you will be able to work with will be 1920x1080 resolution. This is a step below 4K resolution, which by itself isn't a big deal. 1920x1080 (1080p) resolution looks great and works wonderfully. The problem is that you will be converting it from digital HDMI to analog VGA.
But, it is a solution with the cabling that is there and will give decent results. You will need a HDMI splitter that removes copyright protection...
What To Know In this article, I will show you how to bypass HDCP with an HDMI splitter and also list out the top 5 best HDMI splitters for this job. Let’s get started! Disclaimer: We do not support…
pointerclicker.com
I've used the unit by ViewHD before and it works well for 1080p video when I've had issues.
Then you need a HDMI to VGA converter.
Buy a C2G Pro HDMI to VGA and Audio Adapter - F/F at CDW.com
www.cdw.com
I have NOT used a HDMI to VGA converter, and you must use an ACTIVE unit. That is, HDMI is digital, VGA is analog, so you can't just use any old HDMI->VGA cable, it must be a plug-in adapter that takes the incoming digital signal then pushes out analog VGA video.
Another option, which I have never used over actual VGA cable, is a HDMI over RGB extender. RGB cabling is what is inside a VGA cable. So, while I've never tried it, if there is no good way to replace the VGA with cat-5e/6 cabling, then I might give this a try:
Buy Gefen HDMI over RGBHV Extender featuring HDMI over Coaxial Cabling, Cable Runs Up to 150', Resolutions Up to 1080p, 2k, 1920 x 1200. Review Gefen null
www.bhphotovideo.com
Those units are VERY discontinued by Gefen, and would need to be purchased via eBay.
and
I used a set of those in an installation where RGBHV cabling was run to a projector without any conduit and no cat cable. It worked well for 1080p video to his projector.
If you DO open things up to run a new cable, then you will want to run at least one fiber/hybrid HDMI cable and TWO or THREE cat-6 cables. Cat-6 cables are SUPER inexpensive and running extras is always a good game plan.