The problem with your first approach is that you are assuming that all of the initial KE has become GPE at the top of the circle. This means that the mass is not moving at the top of the circle. If this were the case, the string would have gone slack before this point, and the mass would no longer be moving in a circle.
You have to do two things here. One is indeed to put together a conservation of energy equation, including the kinetic energy at the top of the circle, and noting that the mass is not moving around the circle at a constant speed. The other is to think about the centripetal force on the mass at the top of the circle; what it consists of, what you can equate this to, and what condition is necessary for the mass to still be moving in a circle. This will give you two equations, which together, should give you the answer.
Sorry this is a bit vague, but you'll gain much more from having another go that you will from seeing a spelt out solution. One clue, though; it doesn't matter what the value of the mass is, the answer will be the same for any mass.