Available resources :
Significant river in the north, a desert in the south, and sea to the east and west. Your northern neighbours might occasionally come and occupy or loot the cities. If they are powerful they might demand tribute from you, if weak they will pay tribute in exchange for a quiet border free of raiding and unrest. Realistically it would vary over time. I guess the normal would be an exchange of gifts based on wealth and power.
In any case your neighbours neither want your land nor have any capability to hold most of it, so your defense is fairly good. Similarly you are extremely unlikely to be attacked from the sea or the desert. Most of your wars would be to suppress uppity tribal chieftains to submit to the power of the king.
Steppes grow cattle and horses. You can export meat and leather from cattle. You will also probably almost always have more and better light cavalry than your neighbours. You can easily raid their lands if you wish while an attempt to harass you beyond the border and the few cities and towns will result in losing whatever army was wasted on the attempt.
A flat desert you describe will probably have evaporites. So you can expect to mine and trade salt and maybe nitrates and gypsum. This could easily be a royal monopoly (nobody actually owns the desert) and give the monarchy an independent source of income. The salt trade would also be a good reason to have a king keeping things in order and suppressing unrest. And the wealth from trade would make it worthwhile for the chieftains to submit to the royal authority. You might also get placer deposits of gold and gem stones.
Your tributary river makes shipping the riches of the desert to markets much easier. From your capital you control the river trade to the north or to the port city.
Actual answers :
How would the king maintain control over his people with such harsh geography?
By controlling mining and trade and then leveraging it to rule the nomadic tribes. Those who support the king get their share of the wealth and get to sell their meat and leather in the markets the king controls. Given that the wealth sharing might happen in the form of steel weapons, I'd expect that loyalists protecting the trade will outcompete rebels trying to loot the trade. Or even tribes trying to just opt-out of the kingdom.
Alliances would also be cemented with marriages, so after a century or two, you can expect leading families of all tribes of any significance to be related to the royal lineage. So the king would have the best connections and likely be the one source of stability needed for the trade.
Monarchies often have some religious component as well. It might be as simple as the first king who unified the tribes having legendary status. It might be a story about the royal family descending from an actual God. In any case the royal family has the status and respect that possible competitors do not.
The king is also the sole focus of the network of alliances of the tribes, so rebellions against him would normally be opposed by other tribes and the largest alliance that can oppose him would normally be much smaller than the one supporting him. This can of course fail if a weak king coincides with a charismatic and ambitious challenger. But the rival would normally be part of the royal family as well, or at least claim he is.
The king is also the sole source of stability and peace, so the traders and usually the northern neighbours would back him against rebels.
How would the kingdom's citizens live, considering the lack of land good for building settlements?
Not sure what you mean with this. Steppe nomads do not really need settlements except for trade and you have that covered. Also the steppes and even the desert will probably have places that are good for agriculture and work as major nodes of commerce. Purely from the description of land I'd expect more cities along the rivers and good spots in the steppes and the desert. I'd assume the king is actively suppressing the formation of cities other than those he needs to control trade. This sounds silly, but is actually a valid strategy to prevent conquest by his northern neighbour. Even the strongest army cannot conquer a city that does not exist. And nomadic tribes are a much better source of cavalry than city people are.
Lastly, how do I justify the kingdom not being invaded?
Natural borders, lots of cavalry, and few locations you can conquer and hold against the said cavalry. Trading for peace and trade goods is clearly a better option as long as the king is powerful enough to keep the peace.
Minor complaint :
If you have a river along the northern border there must be height difference east to west. So you would probably need a coastal mountain range on one side. Of course the mountains can be old enough to be closer to hills if you wish.