The three constants you give illustrate the arbitrariness of units. The magnetic constant

$$K_m = 10^{-7} {N\over A^2}$$

serves to define the Ampere. The definiton of the Ampere implied by this (defined) constant is that the force between two long wires carrying one Ampere of current at a separation of 1 meter is 10^{-7} Newtons per unit length. If you change this constant, you redefine the Ampere.

the other constant you mention is

$$ K_e = 8.996 \times 10^9 {Nm^2\over C^2}$$

This is interesting, because if you just look at the number, forgetting the power of 10, it's the square of the speed of light. The reason for this is that electric and magnetic effects are related by relativity, and the ratio of $K_e$ and $K_m$ is the speed of light squared. Because we choose units where $K_m$ is a power of 10, the $K_e$ is then the same as the square of the speed of light.

Since the speed of light, like $K_m$, is defined, thereby setting the standard definition of the meter, the electric constant $K_e$ is also defined- it is $10^{-7}$ times the exact speed of light squared. If you vary it, you can only do so in conjunction with varying the definition of the speed of light, and therefore the meter.

The **gravitational constant**, in an ideal world, would define the unit of mass. But gravity is too weak to measure accurately enough, so we use a block of metal in a vault in Paris for now to define what a "Kilogram" is. This will probably change at some point. But "G" is also a constant that is philosophically incapable of varying, since it defines the system of units.

If you want to ask a sensible question, you should ask them of quantities that don't depend on the units. In practice, you ignore $G, \hbar, c, k_b, K_m$ (your stuff, swapping out $K_e$ for c, plus Planck's constant and Boltzmann's constant). These you set to 1 in a reasonable system of units, and then all *other* quantities become dimensionless and meaningful, so you can ask about why they have the values that they do.