A lot depends on how much work you want to go. That joint looks like it was designed to fail. The dowels are too big in diameter for the size of the stock. Then it looks like the screw didn't get a large enough pilot screw through the piece that broke.
Repair method 1: Cut two pieces of wood into 45 degree angles, where the right angle sides are at least 2 or 2-1/2 inches long. Predrill loose countersunk pilot holes into the hypotenuse side of each, so that you can drive a No. 6 cabinet screw into the broken part and the rung. The screws will need to far enough from the ends of the rung and broken parts to prevent splitting, so pilot holes suitable for the No. 6 screws will be needed there, also. Sand the joint area where the braces will go with 180 or 220 grit sandpaper and clean off with isopropyl alcohol. Then reassemble with screws of suitable length and clear epoxy. I typically use Permatex 5 minute epoxy for things like this.
Repair method 2. Cut out the broken part so there is about a 6 inch gap. Be certain that the cuts are squarely 90 degrees. Using pine or fir of the same thickness and with, cut a patch the same length. Drill new holes for the existing dowels in the rung and also a pilot hole for the attaching screw. Cut two pieces of pine or fir that are about the same thickness and width to use as splints, each at least 4 or 5 inches long. The splints need to overlap the remaining frame rail by at least 3 inches, and also nestle onto the rung, which may not be a right angle. The ends away from the rung can be beveled so as to not be a toe catcher. Drill loose, countersunk pilot holes for at least one No. 6 cabinet screw at about 1-1/2 inches from each end of each splint. Sand the splint contact area remaining frame with 180 or 222 grit sandpaper and clean off with isopropyl alcohol. Using clamps and clear epoxy glue, affix the patch to the cut out part of the ladder and the splints to the inside of the above and below the rung. Secure patch and splint with No. 6 flathead cabinet screws, after predrilling suitable pilot holes.
Repair method 3: Do as others have suggested, which will be faster and easier, using steel angle plates. Consider bolstering the plates and the broken wood part with epoxy.
I tend to go for excessive and elegant, where if the item breaks again, it breaks someplace else.