Mike -
To do this properly you can't do it the easy way. Many people think taking up the slack on the cable is the easy fix, but this NEVER works as the Healey needs the full swing of the balance lever (the lever attached to the axle housing) to get a good grip on the brakes.
The best procedure is to first take out and inspect all of the pivot pins, make sure none of them are worn. If any are worn, they must be replaced.
Second, make sure the cable adjustment is slackened off enough to where the balance lever connection is at a 45 deg angle BACK from the plane of the axle.
Third, with both rear wheels up off the ground (front wheels chocked!), move the mechanical adjustors until they grip then back off two clicks usually.
Finally, take up any slack in the hand brake cable at this time, but don't take up too much otherwise you'll engage the handbrake.
That should fix it. If it doesn't, and you have an early BN1 with 4 stud axle, you may need to replace the handbrake tappet roller. Thankfully you can buy these new from cape-international.com. They are the only people I know who sell them.
Alan