Is this problem NP-hard?
It seems very likely that it is $NP$-complete.

A good "trick" to tackle a problem like this would be to create an instance where there are $X$ very big numbers in the grid and you're allowed to choose $X$ items. Choose the goal amount so that you need all of these big numbers, and this constrains you to "use" all of the $X$ items to cover the $X$ big numbers. The only freedom is in where exactly you pick the position of the rectangles relative to the big numbers.

A good candidate for a reduction would be Vertex Cover, which is $NP$-complete even for planar, 3-regular graphs. You could fix some embedding of the graph in the plane, and then place large numbers on the edges, so that you need to collect these numbers to win. You'd make $K$ so large that you can cover almost all of the big numbers in each edge, but not quite. Placing a rectangle at the position of a vertex would cover numbers in all of the incident edges.

I'll leave it to you to figure out the details (since you only asked for a "way" to prove it) but I'm pretty sure this is a feasible approach.