The best military wonder is Martial Monastery/Sanctuary. Both of its benefits are important to a military build. While the culture benefit is not great earlier in the game, by the woodelves chapter it is almost higher per square than even the best event buildings. Culture is critical for military because all military building use more culture than population and often 2:1 culture : population. The health benefit is a little obvious, but what is not obvious is that the autofight AI will regularly walk your high initiative troops forward into range of lower initiative opponents giving your enemies the first hit. The health bonus ameliorates that bad choice. I suggest building the Monastery and leveling it to 6 as soon as possible and raising to maximum as opportunities present.
Military productions requires 2 things: supplies and time. Supplies can readily be gained from workshops and PoP enchantments can easily enhance that. That makes time the critical component in production. There are only 24 hours in a day and not too much you can do to change that. Instants and diamonds are better spent elsewhere. This means that wonders which improve the speed and duration of production are the second class of important wonders. Needles is the first of that category and given that it also enhances damage for a unit that dominates one tournament and supports in 4 others, the needles is the most important of this category. It should be upgraded to 6 rapidly and then considered for further upgrades later when both Bulwark and Shrewdy are L6. Next in this category are Shrewdy and Bulwark. Both should be built up equally as you have spare KP and whenever your training queue does not last for continuous 24 hour production. The free units provided by these may be enough to allow you to not have to train any of those unit types, or can free up time to produce Training Grounds and Mercenary Camp units for those weeks that would use the LM and LR unit types heavily. Keep in mind that the training grounds is sometimes faster, but usually slower than the barracks with needles. The final wonder in this category is the Flying Academy. By the time you can build it the mercenary camp troops are only slightly useful, and mages are often ineffective in combat. This wonder can be leveled slower and it is often better to focus on raising the monastery towards L16 until more of the camp units get promotions. If your FA is below L6 once the monastery has gotten to L16 then raise the academy.
The other wonders that increase troop damage are all useful and should be built if you have the space. That space can often be found by demolishing factories (I have had a 1-1-1 build since chapter 5 and keep stockpiling more goods), workshops (I reduced mine from 10 L15 to 5 max level during orcs), and the residences that were used to provide population to those buildings. However leveling those wonders should not be a high priority as the L1 bonus is the biggest single step.
Wonders that provide supplies: Prosperity Towers, Endless Excavation, and Tome of Secrets can all be useful and have ranked usefulness in about that order.
If you happen to have a city on Elcy you can check mine out. Last tournament I did nearly 3000 points and only catered 4 or 5 encounters because of silly loses. I could do that because I started the tournament with 80 squads of golems. I got 160 KP during the tournament and since my fellowship was in recovery mode only 15 more from the chests. In the last hours of the tournament, I could see I would not get any more KP and we were not close to gaining another chest so I stopped for that week and reset my training queue for heavy melee, which is often the go to choice in crystal. I expect to start the crystal tournament tomorrow with 90 squads of treants and that was nearly all trained starting from Saturday at 11am.
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