If one of the three cables is broken, then you have to replace it. But it could be the diirt and corrosion where the cables meet the mirror, or the cables are just stretched. Push on the mirror and see if the mirror is "flopping" in any direction in the housing. If so, a cable is broke. If mirror still feels fairly tight, then try these two tricks that have worked for me over the years.
First, remove the nut that holds the adjuster in the dash and pull the cable loose from the dash. Next, push the mirror all the way up and you will see a philips head screw under the glass that holds it to the housing. Remove that screw and gently pull the glass and cables out enough to spray some WD-40 on the cables and pivot points to lubricate and clean out grease, grime, rust, etc. Reattach glass and check operation.
If mirror is still slow to respond, then "wind" the cable housing by twisting it 2-3 turns and reinstall nut. This will tighten the cables. See if you mirror works okay then.
Sometimes this won't work, but it has worked several times for me in the past.
Tim