The thickness of the anodization doesn't have much of an effect on thermal conductance. Ironically, thickness does have an effect on emittance. Anodizing has a dramatic positive effect on aluminum's emissivity.
Your conductivity numbers aren't wrong, but they don't really translate exactly to the specific situation. The ability of alumina to conduct heat is only one part of a 3-dimensional puzzle. The first thing that must be considered is that heat conducts through matter in two fashions- in-plane and through-plane. It's pretty much as it sounds- in-plane is how it moves through a single mass and through-plane is how it moves between two pieces of material. As I have written about before, you could take two pieces of the identical material and mate them together as closely as possible and they won't conduct the same as the single solid mass.
Heat is the energy of excited electrons. Fourier's law states that heat moves towards cold. In a sense, cold doesn't really exist. Only the lack of heat. If you think of this on the atomic level, the energy of one excited molecule is easily transmitted to the one atomically bonded to it. Think of them as a bunch of spinning tops- as one bumps into the one next to it, it transfers energy. Once it comes to the end of the mass, it can only transmit energy at the rate the material contacting the mass is able to accept. If you do this in a vacuum, the energy will stay in the mass- it has nowhere to go. If you do this in air or water, heat moves at that material's ability to conduct heat. (We're leaving out emissive radiation but that's another story)
This is where conductivity becomes less absolute. Each excited molecule has a certain amount of kinetic energy. When molecularly bonded with another molecule it transfers energy quite well. But that kinetic energy also allows the molecule to transfer energy to another molecule that's very close to it but not atomically bonded. A simplification would be that the energy is able to jump the gap.
In a practical application, the closer these two surfaces are to each other, the higher the level kinetic conductance. As you move these surfaces away from each other, this effect diminishes logarithmically. At the atomic level, the smoothest, flattest surface finish we are presently capable of achieving looks like a field of boulders. Actual molecule-to-molecule contact is a very tiny fraction of a percentage point of total "contact" area. Everything else in between has some other element filling the voids. If you were able to place molecules the same as the mass in those voids, they would atomically bond. So, the closer you can get two surfaces together, the more kinetic conductance you can achieve. This is the reason why the thermal conductance rate of alumina (anodization) isn't absolute in a practical application. Now, if you build a significant layer of alumina on the surface, eventually the kinetic effect is lost and now you are dealing with a material that conducts at a much lower rate.
This kinetic effect is the reason all the manufacturers of thermal products direct the thinnest application as possible. I have seen a few articles where computer cpu coolers were evaluated. In those articles, a rough contact surface was smoothed and cooler efficiency improved.
Plating accomplishes not a whole lot. If you plate a piece of copper with silver, what you have is one very conductive material transferring heat into another, more conductive material very efficiently- but only for a very brief distance. It's like driving to Grandma's and beginning your trip by driving to the end of your driveway at 250 mph. If the rest of your trip is at 35, you accomplished little. The other factor with plating is that it's a chemical bond, so while the conductance is more efficient because of the closer relative distance of molecules, it's not absolute as if identical elements were molecularly bonded. Plating is often used to prevent the oxidization of the base material. As most metals are reactive with oxygen, as soon as their molecules are exposed to oxygen they begin to corrode. But silver doesn't really oxidize any much more differently than copper so there's not much to be gained. If you are plating for the purpose of allowing a diode to be soldered to aluminum, that's alot of effort for not alot of gain.
To return to the practical application of the OP, anodizing aluminum to allow the soldering of a diode is not an efficient approach. Unless a portion of the overall physical design dictates this approach, there are better ways that will conduct heat more efficiently.
Keep in mind, conductivity is a function of how well the material conducts and how much there is of it. Think of it like plumbing- bigger pipe/higher pressure is going to give more output that smaller pipe/lower pressure. You can balance the two in order to achieve your desired effect.
In the application of conducting heat from a diode, you have to consider each and every element in the chain of process. The first thing the diode contacts is the bonding agent. If you glue it, there are bonding agents that conduct between a fraction of a point and (reportedly) up to 60 W/ mK. Solders can range from the mid 20's up into the 70's. If you affixed your diode with 50/50 solder, none of what comes after it matters all that much. Once you're past that very tiny bottleneck of the led's thermal pad, it becomes simpler. If the design that required a very small footprint had significant thermal output, the builder would be much better served to mounting the diode to a copper post and affix it to whatever the base material might be.
I saw a build where the diode was affixed to the top of a small-diameter copper post. Only the thermal pad in the center of the diode was in contact with the sink. The electrical contact pads were over voids which allowed wires to run underneath the diode. This gave away all the thermal transfer ability of those two pads- a significant amount of area. At the same time, this design was conducting heat out and into the wiring. As I'm sure you know, the hotter the material, the more electrically resistive it becomes.
Thermal management requires a holistic approach. You must understand every facet of the design's thermal flow and focus on the bottleneck. Nothing in the design will conduct any more effectively than that point.