Even as there are many ways to "skin a cat," there can be several ways to climb a peak. The following is how we climbed Heisshorn. It may be difficult to follow our exact description and what's provided below may have some errors, but this is our best recollection using the photos we took to help us recall the route. Also, Google Earth is off considerably and shows Heisshorn missing about 300 feet. The summit coordinates we've provided therefore are likely off by a significant amount.
From our suggested best campsite at 11,600 feet, go back to the main trail and follow it up eastward toward a pass at 12,595 ft., NE of the Heisshorn. The trail leads up through the forest for a short while, then opens into a vast, tundra covered area. The lush growth can be quite wet from overnight sprinkles and dew. About half way to the pass, you may depart the trail and hike more directly south toward the north ridge of the Heisshorn. As you approached the ridge, the tundra thins out to a soft dirt. Once on the ridge, walk on up to a flat area right at the 13,000 contour. At this point, the terrain changes abruptly. Below, there is soft dirt, tundra and a few rocks. Above is the rocky morass of the remaining 400 feet to the summit. Pause to arrange things and study your options.
The most obvious route from here we saw was to stay left of the ridge (south side) and follow up what appears to be a faint trail that leads through the piled up rubble of boulders and rocks. It could best be described as a “ramp” of sorts that skirts the ridge for the next two hundred vertical feet. As you ascend, the ramp will gradually disappear so begin more of a scramble over the loose rocks back in the direction of the ridge line that steepens even more. After about 200 feet of gain (at the most),emerge out onto a flat section of the main ridge. For the next fifty yards or so, you can easily stroll along the ridge crest, which becomes something like a narrow catwalk here. From this vantage point, the peak appears like a giant canine piercing the clear morning sky. On the north side, it plummets away precipitously.
On the south side, the ridge angles down a steep slope of precariously perched, loose rocks, too steep to ascend or descend safely. Continue following the ridge and after an open section of ridge and some boulder hoping, drop off on the north side and contour below some short cliffs and great blocks, crossing the head of a gully and emerging at a rocky, tight saddle, at the head of yet another steeply angled couloir on the north side. Ahead will lie the last 200 feet of mountain. So far, so good. But the hardest part is to come. From here, it's about two leads to the summit. A short wall of rock will be in front of you, topped by a pile of broken, loose rocks along the very steep ridge line. Using a rope here may be a good idea for anyone shaken by the loose rock and exposure, but in terrain like this, there's little to use for a solid anchor. Climb up the short wall and then gingerly proceed over all the rubble for 125 feet or so to another area where the ridge flattens again.
Walk through some large blocks of rock and come to yet another head of another couloir and another short, rock wall. This section of large blocks appears as the main break in the ridge line when viewed from the catwalk section below. One more lead separates you from the summit if using rope, which is about 100 feet away. The north face of the peak drops off in sheer fashion here adding to the drama of the ascent. After mounting another rock wall-like section, work carefully over loose rocks and rubble stacked on top of each other. The climbing is barely 4th class, and only for a short move or two. The remainder is 3rd and 2nd class, but the exposure is great. As the angle of this pitch relents, walk the last few feet to the small, rocky summit.
For a descent, we recommend trying to go back down staying as much as you can on the route you took up. Staying on familiar terrain is usually safer since you may have previously identified the worst hazards. Make your way back to the 12,595 pass where the trail crosses over to the East Fork of the Cimarron and decide if you've had enough or want to continue north to El Punto (UN13,319).
Links to other information, routes & trip reports for this peak that may be helpful.