Short answer. The relation between Newtons and kilograms, with respect to work / kinetic energy, is actually just through Newton's Second Law!
Here we are not mixing up Newtons and kilograms (forces and masses). The mass plays a role in the kinetic energy equation precisely because mass plays a role in how much force it takes to accelerate an object from rest to a speed v.
Example.
Consider for instance an object which has a mass of, say, 0.1019 kg. The gravitational force that this body experiences at the surface of the Earth is −1 N (that is, a downward force), if we take g = −9.81 m/s². If we hold this object from a height of one meter, and allow it to drop, gravity performs work on this object, exerting a force of −1 N over a (downward) displacement of −1 m. As a result, just before impacting the ground, the object will have 1 Joule of kinetic energy, as (−1 N) · (−1 m) = +1 J; gravity will have done 1 Joule worth of work on the object, which causes it to move with 1 Joule worth of kinetic energy.
In the examples you give, the correspondence that you're looking for is between mass and the gravitational force that it exerts. When you say that you take "1 Newton" and lift it one meter, the 'Newton' you're referring is one Newton of force exerted downward by an object with some mass — or more to the point, the one Newton which would be the minimum necessary to raise it steadily by opposing gravity.
Derivation
We can actually show directly how mass comes into the kinetic energy formula, and pinpoint the reason that it's there.
Suppose that we exert a net force F on an object with mass m, over some displacement d in the same direction as the force, where the object starts at rest. The amount of work done on the object will be
$$ W \;=\; \mathbf F \cdot \mathbf d \;=\; Fd,$$
taking F to be the magnitude of the force F, and d to be the length of the displacement d.
Because we're doing net work on the object starting from rest, this work will go directly towards the kinertic energy of the object.
The acceleration of the object under this force is
$$ \mathbf a \;=\; \mathbf F / m . $$
If t is the time that it takes for the object to be moved by a displacement of d, we have
$$ \mathbf d \;=\; \tfrac{1}{2}\mathbf a \, t^2 \,;\qquad\implies\qquad t \;=\; \sqrt{2d/a\;} \;=\; \sqrt{2dm/F\;}.$$
taking the magnitude a = || a ||. The speed that the object is travelling after that amount of time is just
$$\begin{align*}
v \;=\; a t \;&=\; \bigl(F/m\bigr) \sqrt{2dm/F\;} \\[1ex]&= \sqrt{2Fd/m\;}, \end{align*}$$
applying Newton's Second Law again for the acceleration, and cancelling factors of F and m under the square-root.We may then re-express this equation as
$$ v^2 \;=\; 2Fd/m \qquad\implies\qquad \tfrac{1}{2}m\,v^2 = Fd = W, $$
where we just applied the formula for the work done on the object at the end. Because the net work is the same as the kinetic energy in this case, it follows that $K = \tfrac{1}{2}m\,v^2$.
In the derivation above, the only ways we used mass was with Newton's Law, a = F/m. So, the fact that it occurs in the formula for kinetic energy isn't because we are confusing mass with force, but because of the relationship between mass and force.
