Alright, let me give recommendations in order...
Junkojuro: Very good! Though we'll need to come up with a cover story for him. Takeshi while in recent years involving himself more in spiritual manners, meditation and the such is blind to the supernatural world and while devout, he acts more out of faith rather than actually having signs of the spiritual world being quite palpable.
Hana: Very interesting character as I expected! I quite like Hana, however, I feel like she needs to be toned down a little. At the moment the Heir of the Mitsumine is young and barely coming into his responsibilities... thus him and by extension, his entourage have a lot of leveling up to do. And as she stands, Hana seems like she's already the ultimate version of what she is. I do want her to have the talent and promise to be one of the greatest economist of the realm, however, I don't want her to already be there by the beginning.
Kumori: Will reserve notes on him until the application is done, however, I shall reveal some plans for ninja that I had and that I unfortunately, didn't share before this moment. There are about three major ninja clans hidden in the nation. These pledge loyalty, in general, to not one house... instead, they operate in their hidden forts, take contracts by pay... but they do build relationships with one family or the other. For instance, Boar, Wolf and Turtle may all share their same alliance to a particular shinobi school who by the right price provide them with their members to reside around them as spy masters, bodyguards and general advisors, with possibilities to hire strikes on particular targets.
As for the questions... Economically, the Mitsumine's are doing just fine if perhaps not on an optimal position due to oversight on Takeshi's part due to him retreating to his manse. The branch family ruling the capital and the mines are squandering their wealth after securing their holds for a few years now. Still, economy is currently stable and their is a comfortable cushion of coin in the treasury to handle an emergency or start a new project. Likewise, there are stores of rice with enough food in them to feed roughly 25% of the population during a winter sorely in reserve. However, something to be said of the mines is that they are not developed to the max. These could be expanded, incorporate newer technologies and more sites with ores could be found and worked on.
As for their armies... the Mitsumine currently don't have a standing army due to being in a time of peace. The keep on the passes to the Mitsumine lands is currently unmanned and abandoned for 9 years. Instead, a small but highly mobile force of 'rangers' keep watch on the passes and the lands beyond (The Mitsumine have roughly three or so farming villages beyond the pass to the majority of their lands). All of the cities have their own city watch and as usual, the nobles of the Mitsumine tend to all be trained samurai when possible. There may be a couple families of warriors that still train in their ways and keep themselves sharp.