This is my two cents on it. It's been a while since i did physics like this, so I'm not up to speed on formulas etc.
1. Rotational velocity is irrelevant in theory. The fact that the earth does spin may be seen as the fact that the universe spins about the earth - perfectly reasonable. One might argue that the sun in what the universe spins about - the sun is merely just another star. True, the planets in our solar system moves around the sun, but then again our solar system is only a part of a galaxy, which is part of space, and there is relative movement everywhere.
2. Let's first make som statements:
a) The further you are away from Earth, the less the force of gravity becomes, but at no point will the force of gravity become zero.
b) Because it takes work (i.e. an addition of energy) to escape the Earth's gravity field, the potential energy must increase when distance is increased.
c) The datum from potential energy may be placed anywhere. An object may be above or below it.
3. Then let's think about those for a while.
d) Because of a) and b) the rate at which the potetial energy increases, decreases as distance increases. This will then be a function which tends to increase to a certain specific number as it's operator, the distance from earths, tends to infinity. It will never reach this number however (if this puzzles you, read on convergent sequences). This matches our second part of the statement a).
e) Because of d) and c), we will put the datum for the potetional energy, Ep=0, at this number. As our "normal" datum usually refers to the surface or earth, this datum will refer to the position where an object would stand still, and not be affected by Earth's gravity. Referring to a), this would be imaginary. However, and this is the biggie, if we can give an object more total energy (Ek+Ep) than this number, it will be able to escape gravity.
f) Newton's Law of Gravity implies that Ep = -(GMm/r), where G is a universal gravitational constant, M is the mass of the Earth in this case, m is the mass of the propelled object, and r is the distance between the two. We know that Kinetic and Potential Energy is interchangable in an ideal system, so Ek would have to be at least equal to than Ep. I.e. Ek=Ep, or (1/2)mv^2 = -(GMm/r). We rearrange the formula to (1/2)mv^2 - GMm/r = 0. Further we can say that m((1/2)v^2 - GM/r)=0, so the mass of the propelled object is irrelevant. By solving for v, we see that v=(2GM/r)^(1/2).
This shows us that the escape speed is dependant upon the following: the mass of the body to escape and the distance between the two objects' centers of gravity at start.
With G=6,67E-11, the earth mass= 5,97E24kg and the earth's radius at equator r=6378E3m, we are able to compute the escape velocity: type in (2*6,67E-11*5,97E24/6378E3)^(1/2) into google and see for yourself. The answer comes out to approximately 11 km/s.
3.
The key is to have a rotational velocity, which in turn will give you the acceleration to overcome gravity.
You are implying that a (rotational) velocity will produce acceleration. If this acceleration is other than gravity, you will have to add energy somehow - speed does not add energy by itself. And for gravity, that's what we are trying to escape, right?
4.
you can assure your daughters that nobody knows how gravity works
Very true indeed, and a good point to make! We do however need to keep in mind that even though we do not know why gravity works, we do know, fairly accurate, how it behaves. These physics are well-researched, and the math appratus around this has been there for a long time.
5. Kudos to Genghis for a, as i now see, much simpler explanation
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About convergent sequences: Try to add 1 + 0,5 + 0,25 0,125.... You can go on as long as you want to, without ever reaching 2. The sum of the sequence is always increasing, but will only reach 2 in infinity, which of course never exists in practice. Still confused? Find a 2-meter rope. Cut it in half. Cut one half in half. Take the one quarter you get and cut that in half. Go on, make as many cuts you can. How many cuts does it take before all your pieces is exactly two meters, again?