You seem to have a severe lack of understanding of what the Romans were capable of, and the scale of things 2000 years ago.
Logistics isn't what it used to be. The storage and preservation of food isn't what it used to be. Hygiene most definitely isn't what it used to be.
Food and hygiene
Packing that many people in such a small area in medieval times will result in starvation, and quite probably outbreaks of terrible diseases.
Agricultural productivity in medieval times was quite abysmal. Crops would fail all the time. Entire herds of animals would get sick and die. Thousands of people would die in a famine, and it would be no more than a sad event in a long list of similar ones. Disease was rampant, and only a small percentage of children - the strongest, and luckiest - would make it to adulthood, not to mention the many adults who would also die from something as simple as the flu.
You're packing a lot of people in a very small area. Sure, we do it today, but that's because we import food from half the world away via trains, trucks, etc. There are thousands of trucks driving into any given large city at any given time of day. If those food shipments stopped for even one day, you would notice shortages immediately. No more fresh bread. No more fresh fruit. Canned food stuffs would start flying off the shelves. We live in a very delicate balance in the big city.
Your 1.75 million people will be in bad shape in such a small area.
Your army and defenses
A wall 50 meters tall, 14 meters wide, and 156 km long is not just incredibly difficult to build, it's also a terrible defensive structure. You've essentially built yourself a fancy cage, and don't even know it yet. A wall needs to be constantly maintained (financial drain), as well as manned. Not to mention, that people need to get out, so you'll have lots of gates, which means weak points which can fall to the enemy.
Now on to your army. A modern country can only maintain an army around the size of 10% of the population before it starts running into economic woes. In war times it will rarely increase above 15%. Not sure how you're going to feed and equip 112,000 people, as these numbers were not encountered in medieval times. Some of the most famous battles of medieval Europe were fought with 5000 people or less on each side.
Even assuming you can keep your people fed and healthy however ...
You will lose
The Romans were the very embodiment of "improvise, adapt, and overcome". Their armies were not always larger than their enemies, but they were highly trained, highly organized, and well equipped. They were renowned for upgrading their armor, shields, and weapon designs based on tips and tricks picked up from conquered or enemy nations.
Your main weapon - crossbows - will only help you as long as their legions are further than arms length away. Furthermore, once the Romans come in contact with your troops and capture some of those weapons they will very quickly end up being used against you, and their armor will be improved to provide better resistance against them.
But I don't think this will even happen until after your nation has fallen. Here's how I think it would all go down:
The Romans will scout your walls, see where your defenders are more lightly concentrated. They will look for a weakly guarded gate, or a place where they can more easily approach the wall with ladders and remain undetected (perhaps due to a nearby forest)
They will then strike, and own your wall before you even know what's happened. Their concentrated and well trained troops will then hold what they've got while their comrades storm through the gate, or over the wall, and establish a bridge head.
Once their troops are assembled and in formation within your perimeter the war is essentially over. They will advance in formation, behind a shield wall, close with your defenders, and slaughter them. A few legions fighting in a determined and organized fashion will wipe out an undisciplined force 10 times bigger without issues.
Note: it's important to understand that a force will lose cohesion, and people will start running away long before most of them are dead. It's estimated that in medieval battles fewer than 30% of the troops would die on the battlefield. The rest would simply break ranks and run away when things started going south. Romans are a notable exception because their forces were highly disciplined troops, trained to fight together, and obey their chain of command. That sort of discipline was essentially unique before them, and wasn't really encountered again until much, much later.
I doubt a long siege with bombardments, and fancy siege machines would even take place.