There may still be tax advantages to installing double glazing, not to mention energy savings, but I don't know the details - you may have to do the entire house. They also may not do much good unless your walls are insulated as well. Are you thinking of doing the whole house or just the bedroom? If the latter, storm windows, or removable interior frames with plastic may be as effective. What stops noise (and heat transfer) is dead air space - not the glass or other barrier material. Another option is to have upholstery foam cut to friction-fit the inside of the window frames that can be put in at night and taken out in the morning (more sound reduction than double-glazing).
For a while, when I had to have the TV on to distract me from the pain enough to fall asleep, my wife used foam earplugs - very
effective for sleeping.
Is it just the incidental noise that causes pain? White noise can also block incidental noise, and for some is less noticable. In the heat of summer we often have a fan running all night, which effectively produces a white noise that drowns out the incidental noises (even the railroad near the house). Now we both need cpaps for apnea; the white noise produced between the machine and mask is very subtle, but does muffle some incidental noise.
All-in all, I think the simple foam earplugs would acheive both the greatest sound reduction and be the least expensive/invasive thing to try. You can get them at any drug/discount store for pennies a pr. and they'll cut noise by much more than double-glazed windows. There are other types that cut more or less noise depending on their intended purpose: