This route is an up-and-back using a rock chute on the steep east side of Ingólfsfjall. It starts out at the Alvíðra farm, at a bit below 100m in elevation See the mountain page for how to get there.
The route consists of two parts, an ascent up a rock chute and then an easy cross-country walk for about 2.5-3 km up a gentle slope to the summit. The walk can’t be more than about 100-150m in gain, so the chute part of the route must be about 350m in elevation gain. (Without a GPS these are only rough estimates, I’m afraid.)
The posts are spaced about every 50-100 meters on the chute when they are most helpful, and are spaced every 5-10 meters on the flat top of Ingólfsfjall, when they are not really necessary. You should be able to see the next post from each post, except at the very bottom of the trail.
From the Alvíðra parking lot, cross the meadow straight ahead of you until you reach the stone wall. Though the map in the parking lot would suggest otherwise, the climbing route starts immediately from this point. You’ll see a small hillock on the slope ahead of you - - go up the left side and then traverse the top of the hillock. From there you should be able to find the first blue post.
You traverse, ascending slightly, across a small line of rock fall and then continue to traverse somewhat more steeply into the main chute. Most of the ascent follows the chute straight up between two lines of rock fall. You’ll find yourself swinging left and right in switchbacks 5-10m long as opposed to going straight up.
Toward the top of the chute the two rock falls merge, and the route swings to the right outside them.
At that point there is about 25m of annoying scree lying atop sand on a very steep slope. There’s a clear line to traverse the scree, ascending moderately to a chain attached to the left-hand side of the chute. The cable isn’t really necessary - - you’re more likely to slip on the scree than on the cabled stretch. From there it’s a short hop up to a cairn at the top of the chute. It turns out that the top of the chute is a confluence point, at 64º N, 21º W.
From the top of the chute, the overly well-marked route begins across the gently sloped flat top of Ingólfsfjall. There’s no real trail, but it’s as easy as walking along Trail Ridge in RMNP.
The trail leads to dead-end at an electrical fence, but there’s a gate clearly visible about 10m to the right of the nominal route. The summit is well marked on a hill in front of you at that point.
Also, I knocked loose 4-5 rocks in the baseball-cantaloupe size range. Because I was the only person on the mountain, it didn't matter, but larger groups may want helmets.
Alternative routes (not tried)
The alternative route presumably starts on the west or northwest side. The west side has a much more gradual slope, and there are several farms on its base. If you wanted to find a west route, I’d look there.
There is a marked trail on the bottom of the east side that I explored for quite a while, and it seems to go around the mountain on the north, where the map shows a dirt track going over a low saddle on the northwest side. There might be an alternative route on that saddle, which I didn’t explore.
So: it seems that you could take an easier out-and-back route to the summit out of the farms on the west side. You could probably also make a loop from Alvíðra by ascending the east route and then finding a route down the west or northwest side to the dirt track. From there, you’d go clockwise around the bottom of the mountain and hook up with the trail on the base of the east side. Again: I haven’t tried this and these alternatives are purely speculative.