You can fix a broken sash cord by using new cord. but to make this kind of repair a one-time-only effort, replace broken sash cords with gash chain.
Removing The Old/Broken Sash Cord
On the side of the window with the broken cord, use a sharp razor blade or utility knife to cut the paint seal between the window frame and the inside stop molding that holds the sash in place.
Being careful not to gouge the stop or the window frame, pry the stop strip away from the frame with a putty knife and remove it.
The stop strip must be removed for access to either the top or the bottom sash: if you’re replacing a cord in the upper sash, remove the lower sash and the parting strip-the piece of wood between the upper and lower Sashes-first
The sash cord is attached to the sash at a groove or indentation in the upper corner.
Pull the side of the sash toward you out of the window frame, just tar enough to expose this indentation.
Pull the knotted end of the cord out of the groove, or untie the knot.
Remove the cord from the sash frame and knot the end
Lift the window carefully out of its track on the other side, you may need someone to help hold it in place.
Untie the cord on that side and remove it from the sash frame, but do not release it-knot the end of the cord first, or it will slide into the wall. Remove the sash from the window frame and set it aside.
If you’d prefer a video on installing sash chains, this one may be helpful:
How Sash Cords Work
Sash cords operate on pulleys, partly visible at the sides of the window frame, the weight of the window sash is balanced by weights that hang inside the frame. To replace the sash cord, you must be able to reach these weights.
Opening The Sash Access Plates
Some windows have access plates in the sides of the sash frame Look for those plates, they may have been painted over.
If you can see only a vague outline, tap the track with a hammer until the edge of the plate is clear.
Cut along the edge with a razor blade or utility knife, then remove the screws in the plate and lift it out. In very old houses, there are usually no access plates. In this case, you must remove the entire inside frame of the window.
Use a putty knife to pry out the sides of the frame.
Install The New Sash Chain
Remove the weight from the inside of the frame and untie the sash cord. Using the two broken pieces of the old cord, measure the length of sash chain needed to rehang the window.
Cut the chain to the correct length, allowing several extra inches so it can be looped through the sash weight and fastened.
Attach a small weight to one end of the chain anything small enough to feed through the pulley will do, Push the weighted end of the chain in over the pulley at the top of the window frame.
Feed in the chain until it is visible in the access plate at the bottom of the frame, then remove the small weight
Now attach the sash weight to the chain. Loop the chain through the weight and bind the end to the chain with sturdy wire using pliers to pull it tight.
The chain must be securely fastened or it will not hold.
Replace the weight in the access hole and reel in any stack in the chain.
Do not replace the access plate or inside window frame yet.
Follow the same procedure to replace the sash cord on the other side with chain.
Set the window sash back into place at the edge of the window frame-you may need help again at this point. Attach the chain to one side of the sash, using a wood screw to hold it in place in the slot.
Lift the window sash back into its track
Attach the chain to the other side of the sash and raise the window (bottom sash) to its highest position, bracing it against the parting stop.
Check the position of the sash weight, the bottom of the weight should be about 3 inches above the sill, Il the weight is not in the right place, adjust the sash chain at both sides of the sash to correct its position
Finally, when the weights on both sides of the sash are properly placed and the sash hangs evenly, replace the access plate or inside window frame.
If the window is an upper sash, replace the parting strip, the lower sash, and the stop strip, if it is a lower sash, simply replace the stop strip.