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.
GEFEN EXT-HDMI-5BNC R HDMI OVER RGB EXTENDER RECEIVER. We want you to be happy above all other considerations. Appliance parts. The value of an item greatly contributes to the amount or effort put into testing.
www.ebay.com
and
Gefen EXT-HDMI-5BNC HDMI Over RGB S Extender. Tested and works great. Does have normal wear from previous use including scuffs and scratches. Includes only the Extender, No power adapter or accessories. Will ship same or next day!
www.ebay.com
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.