Electron energy levels and standing waves problem

I appreciate that at A-level standard, electron standing waves around nuclei aren't explored in great depth, but I'm struggling with the idea of the potential energy around a nucleus forming 'walls' which electrons form standing waves between.
Decreasing an electron's wavelength increases it's kinetic energy proportional to 1/(wavelength)² , and getting closer to the nucleus of an atom makes the negative potential energy more negative, proportional to 1/radius. My study book states that 'if the kinetic energy is larger than the negative potential energy, the electron will escape the nucleus'.
Do the standing waves with longer wavelengths exist further from the nucleus? This would mean their potential energy would be less negative, and closer to their kinetic energy, but I was under the impression (from previous studying) that the electrons with the most kinetic energy were furthest from the nucleus, which explains why they were easiest to remove.
Surely if the electrons have more kinetic energy they exist further from the nucleus, because it takes an increase in energy to get from the nucleus to a point further away!
Sorry if I'm being dim, any answers or suggestions very welcome.
Thanks

Consider that an electron can behave as a wavicle with allowed energy states. We can first find the allowed energy states:

consider that angular momentum is quantised: L=mvr=nh/2pi

We can now find allowed radii by substituting this into the classical expressions for centripetal force and electrostatic force to find the allowed radii:

mv^2/r=kq^2/r^2, v=nh/2mrpi

so solving for r: r=2epsilon0.h/mq^2=an^2 where a is the Bohr radius ~0.53 Angstroms.

we can then use this and our classical expressions for energy to find the allowed energy states of the electron;

Ke=0.5mv^2=1/8epsilon0.r.pi and EPE=kq^2/r (k=1/4epsilon0pi)

so E=(kq^2)/2r

we can then subsitute for out allowed values of r to find the permitted energy states in which the electron can exist.

We can then apply our allowed energy states to the de Broglie relation with which you are familiar and what we find is that the circumference of the orbit must be a whole number of de Broglie wavelengths.

I hope from this you can glean some of your answers, sorry about the lack of LaTex...

EDIT: forgot to mention that for the expressions for the allowed radii and energies the 'n' must be an integer and must be the same - an energy will exist at a radius

Hey thanks for the reply! I'm sorry but I think you're overestimating how much I know... what does:
r=2epsilon0.h/mq^2=an^2
-mean? I've literally no idea, I solved for 'r' using the formulae you gave but I didn't get that at all.
If you could clear that up for me I reckon I could get the rest, thanks for the help!

I remembered the stuff thanks, but I still can't rearrange to get 'r'. I seem to end up with:
(Epsilon*n^2*h^2)/(pi*m*q^2)
I've checked it several times and that's what it comes out as. I don't suppose you could briefly walk me through the maths?