However I'm not sure I'm on the right track here... I'm conscious that the question may require specific formulae for particle physics, and I have no idea where a formulae using the total energy can be found for this question..

Edit... in fact I'm virtually certain this is wrong, the text makes it clear that this is for LOW energy particles, not high. Back to square 1.

Start with the energy-momentum relationship [tex]E^2 = p^2c^2 + (m_0c^2)^2[/tex] where pc is the momentum term, [tex]m_0c^2[/tex] is the rest energy, and E is the total energy. You also know that the total energy is the sum of the kinetic energy and the rest energy. With some algebra, you can use these to find pc in terms of kinetic energy.

You somehow lost the squares on the terms in the energy-momentum equation in your first steps when you worked out your expression for the muon.
I think your equation for the kaon is correct, so just fix the muon one.

This does not follow from the equation that hage567 gave you. In effect, you're trying to go from [itex]c^2 = a^2 + b^2[/itex] to [itex]c = a + b[/itex] which doesn't work because [itex]\sqrt{a^2 + b^2} \ne a + b[/itex].