I understand that you may be in a situation where nobody in yur area can regrind this thing for you. That video makes it look easy. But from years of experience, I can tell you that it is not. The taper in the video is a 40 yours is a 4mt. Little or no room for error on yours. If you must try it, PLEASE,PLEASE, PLEASE, make sure the grinder is parallel with the centerline of the spindle. Any deviation from that and you have a junk spindle when you are done. Any backlash in the axis will cause the angle to be wrong. The taper will only be as good as the machine is. And in your case the machine does not sound too good. If you must, you must. But again regrinding tapers in the machine is all that I have done for over 15 years. I've done over 4000 tapers and would never consider doing any taper by programming the the machine. Way too much room for error.
Good luck to you.
Rocky
www.spindlegrinding.com