What balances out the Normal force in this scenario of uniform circular motion?

There is none; the free-body diagram is wrong. In actuality, what will happen is that the rope will make a small angle but non-zero angle with the horizontal, so that there is some vertical component of the tension that cancels out gravity. The remaining horizontal component of the force will provide the needed centripetal acceleration.

It is not too hard to show that the faster the object goes in the circle, the closer the angle will be to the horizontal. In the present case, with the given values of $v$, $R$, and $g$, it can be shown that the angle the rope makes with the horizontal is less than 5° (try working out the precise angle yourself!) This means that it's a pretty good approximation, as far as the tension is concerned, to treat the rope as though it's horizontal; and if you solved the problem this way, you'd end up an answer that's pretty close to the answer you'd get using the correct free-body diagram. But of course, that doesn't make the given free-body diagram right.