So for our campaign, we're trying something different just to see how it works out. We have a party of several standard D&D characters, and also, a king character.
The king character is intentionally weaker than the rest: to the tune of 5 levels or so, with the rest of the party at level 8. The trick is, his power is in influence and money. He can command armies and make whatever decision he wishes. In this case, the king is an action king that does a bit of adventuring. However, since he has so much power, it wouldn't be beyond the pale for him to decide to take 100+ men from his honor guard with him. What would be the best way to mitigate the advantage that these soldiers would give him in a combat? Having an army vs army situation is one thing, but what about smaller fights where you might have say, 5 – 10 bandits to deal with? The soldiers will quite handily mop the floor with them.
I've already thought of larger monster fights where they're scenery basically, and the PCs do all the work, as well as the cost that toting these men around would add more weight to the already strained treasury.
Clarification: The party members are advisor level, and hold high offices in the kingdom, and are not just adventurers.
I want him to be able to take his retainers with him, but prevent every encounter from turning into "I send my 100 retainers, the bandits die" essentially.
Best Answer
Disclaimer: a large amount of the information in this answer is second-hand or generalizing from groups that wound up having a similar structure as you propose although not entirely by design. Pinches of salt all around.
It's not going to be easy
So, you have one player character who is far weaker than the other characters in combat, but wields a compensating amount of money, influence and soldiers to compensate. I foresee issues. Let's begin with the issues faced by the player of the king.
Combat is a major part of most DnD campaigns, and also where the level difference between characters is most keenly felt. Being five levels behind is a serious drawback, even in the bounded accuracy world of DnD 5e - it's a whole tier difference. While many players can deal with their characters being less useful for a few rounds, a single encounter or a single session, being constantly worse than the others can and probably will make combat seem like a chore for the player of the king. There's also encounter balance to consider - if the combats are level-appropriate for the rest of the party, the king character can quite easily get snuffed by a few area-of-effect attacks not even aimed directly at him.
So, having little power in combat is a drawback, but it can be made up with non-combat stuff like having overflowing coffers, enough soldiers to get rid of mundane threats with ease and enough influence to open the doors to anywhere, right? The difficulty here is that this can potentially detract from the experience of the other party members.
Picture the situation: the party is about to enter a fortified town that's guarded by mercenaries who refuse to let them in. Without the king, the party would have several natural choices: try to sneak around, fight the mercenaries, try to buy them off, ambush a small group of them and steal their clothes for disguises... but with the king, all the sensible solutions revolve around him: using his personal guard to round up the mercenaries, drawing from his limitless coffers to bribe them or simply appealing to his royal mandate.
The core of the issue with such a character is that it's dividing the fun bits of the game between two sub-parties - a problem commonly faced by other "skill monkey" or "social rogue" types in DnD. You have combats that are punishingly hard for the king, and to compensate, non-combat that's dominated by the king, and therefore at every moment of the game there is someone who is not getting to really enjoy the strengths of their character. You may well see people being disappointed, for instance, when the king successfully avoids combat, or conversely see the king's player get disappointed when combat is the only option.
What can you do to make it better?
Avoid building long sections of the game that emphasize some characters over others. A single short scene where the king can pull strings to ensure something nice for the party is nice, but a long scene of court negotiations while the adventurers stand leaning against the wall drumming their fingers is not very nice. Try to focus on scenes where both the party's adventurous skills and the king's wealth and influence come in handy - don't let the king's status solve all problems.
Since combat is a large part of many games, and can last for hours depending on the size of the encounters, it's rather hard to work the king in with their weakened combat abilities. Here, I would recommend a trick I've seen used a few times to give players whose usual characters were absent a bit of combat agency: instead or in addition to the king going to combat, let the king's player send two or three soldiers (personal bodyguards, champions, whatever) into combat and control them. These soldiers should be relatively simple to keep their turns fast, and to not overshadow the regular PCs. The king's player can control the champions so they have more role in combat while keeping the king relatively vulnerable as was your original intent.
Finally, you can simply make the king as powerful in combat as everyone else - it wasn't uncommon for monarchs to be capable warriors as well, doubly so in the world of fantasy. In that case, you would need to bring in the asymmetry from somewhere else, eg. by having enemies recognize and prefer to target the king in combat.