hm... considering the fact that the two nations are at peace with each other, and consider the mountains a natural barrier, I think it's reasonable to assume that niether of them have particularly strong armies. this is reinforced by the fact that the princess was kidnapped so easily from the kingdom that wouldn't necessarily let it happen based on how much like a fairy tale it is. considering this, any remotely sizeable army could probably take the valley, especially if they use a series of scare tactics.

unfortunately, I'm not seeing any major waterways to take control of to allow for easier mobility or creation of supply lines to the front. on the other hand, it means that there are few to no major natural obstacles to overcome once your forces have crossed the mountains. in addition, the fact that the majority of the valley isn't heavily forested suggests that cavalry would be quite effective, especially taking into account the likely difference in numbers and training.

The biggest obstacle, frankly, is crossing the mountains, especially since there don't seem to be any well tread passes through them. the key is finding a pass, flying over them, or digging a tunnel through them. I'd recommend drilling through them, if possible, because you'll be able to move troops through faster than through a pass, and you'd lose less men to exposure. also, steadier footing for the horses.