If you want to understand this, instead of thinking of a transmission line as two wires with special behavior, you need to think of a transmission line as a guiding structure for electromagnetic waves. When we talk about currents and voltages and capacitance and inductance in a transmission line, that's a useful simplification for many cases. But to understand skin effect you should instead think about an electromagnetic wave that is guided by conductive structures. The currents and voltages that we measure on the transmission line are caused by the EM wave travelling along it, not the other way around.

So then, when an EM wave encounters a conductive material, what happens? The EM wave is reflected. It doesn't penetrate deeply into the conductor. But if the conductivity is not infinite, it does penetrate slightly. And up to the depth that it penetrates, it causes currents to flow in the conductor. That is the skin effect.

I'm slightly confused as to the co-ordinates of how you're defining your transmission line.

Normally we call the axial direction (along the length of the transmission line) the z-dimension. The transverse dimensions are x and y.

It's easiest to visualize in a parallel plate waveguide. There, the propagating mode can be seen as a sum of plane waves reflecting back and forth between the two plates. So these plane waves have a k-vector that has components in both the z and x directions. The reflections off the conductors is what causes the currents in the conductors, and the depth of penetration into the conductors is what determines the skin depth.

Inductance:

The above is the formula for the inductance of parallel wires where s is distance between wires, L is the overall length and d is wire diameter and here is a calculator that uses the formula.

Capacitance between two parallel wires is:

For series resistance you will find many references on google for this and please note that it is ambient temperature dependant. There are plenty of references for skin effect also but not so many for the formula for proximity effect but hopefully this reference from wiki might prove useful.

Finally, once you have the resistance, inductance and capacitance you can use the standard formula for characteristic impedance (applies to any length of wire of course and is still valid down to DC): -

You haven't considered "G" but this is usually a very large resistance and its effect is negligible.

So now you have the impedance looking into an infitely long line of cable. However, you want the impedance looking into 100 metres of such cable so you will then have to work with reflection coefficients and the load impedance (stated as unloaded in the question) BUT YOU CANNOT ignore these load effects (and reflections) if you want a proper answer.

Reflections will occur with any incorrectly terminated cable: -

And these can effect results significantly even at moderate frequencies like 100 kHz. At 100 kHz the wavelength is 3 km but this is probably going to be more like 2 km. Major things happen at a quarter of this wavelength i.e. an open circuit can look like a short circuit and a short circuit can look like an open circuit so 500 metres is still pretty close for creating major errors when considering a cable that is 100 metres long.