"It seems I forgot the keys."
Translation:Кажется, я забыла ключи.
It makes little sense with «что»—a structure without мне but with что inluded would mostly likely be used if you did't forget the keys but it only looks as though you did (somehow). Quite a good interpretation in sentences like “It seems as though the whole world is against me”; not so much here.
Yeah, мне кажется would work. However, when you use just «Кажется» as a cliché sentence starter, it is rarely used with что when you really mean "I think"—but it IS used with что when you mean a more metaphorical "It seems almost as if..."
And let us not forget about the past: you can use «Казалось, (что)» in that second meaning (comparison) to tell stories about what events in the past seemed like.