Right... how I'd do it, right or wrong! I assume you can't get the info online or from the folks at Enco?
Start out by getting a couple of socket-head (allen) screws that fit in the threads on your existing chuck, screw 'em in and measure across the outside of the unthreaded bit, note that measurement down, measure the inside, note that down.
Do this on all 3 hole-to-hole sides of the triangle, add 'em up and divide by three to get the average inside and outside measurements, then add inside and outside and divide by2 to get the centre-to-centre distance (the side of the triangle) - the averaging is 'cos the hole centres may not be accurate

Usually this'll give very close to a Sensible Number, like 120mm, 95mm or similar...
Once you have the centre-to-centre distance, divide by the square root of 3 to get the pitch radius, double that for the diameter and Robert's your parent's sibling! If you have a hole in your piece of plate, make a "spud" to fit in it, and *while still chucked from turning* run in a centre drill to give somewhere for you dividers to pivot while you mark the PCD.
To get the taper angle right, measure height of the stub *accurately* (height gauge, sticky-out-bit on digi calipers etc.), and the difference between the fat and thin ends of the taper - the difference divided by the height will give the tan of the *included* angle of the taper, you need to set the topslide around through half this angle to get close to the taper.
Once you've set the topslide, make a taper on a piece of scrap, longer than you think you need, with an undersize little end - try it for fit, then face off the little end until you get a snug fit in the back of the chuck you already have - then it's marking blue time, to prove whether the angle's spot on, adjusting the topslide if needed, facing off the small end if it's too small.
Once you have it right, keep this plug gauge safe! Your topslide's set to the right angle, so leave it where it is and chuck your backplate-to-be and take a facing cut using the cross-slide then bore an undersize recess for the taper using the carriage feed, and putting on the cut (upside down boring bar on the far side of the hole) outwards from the central hole using the topslide.
Once you're close, use the (angled) topslide to cut the taper, fine cuts, testing with marking blue, until you get a good fit. Once it fits, use strips of silver fag paper as 1/2 thou" feeler gauges and take *really* fine cuts (or polish it with emery paper) until the strips are trapped between the spindle nose and the backplate - once you've reached this point, you can bolt it on the right way round, face it, cut the register to fit your new 4-jaw and you're done

Hope that makes sense, and helps rather than confuses!
Dave H.