Yes a powerful enough laser can ionize nitrogen.
Now the question of how powerful the laser needs to be is a tricky question to answer because it depends on how each nitrogen atom (or molecule when in the air for example) is ionized. Ionization mechanisms can be broken down into two common categories; photoionization and collisional ionization.In photoionization, the electron is ejected from the atom by interacting with an electromagnetic field. In collisional ionization, an electron is ejected when a particle, for example a free electron, collides with the atom.
Photoionization can occur in many different ways. The wavelength requirement of $88$nm is for single photon ionization. If we convert 88nm to a photon energy, $\hbar\omega$, then we get $\hbar\omega=2\pi\hbar c/\lambda = 14\text{eV}$. This is about the ionization energy for nitrogen $14.53414\text{eV}$ (see the CRC value on wikipedia:ionization energy). In this process, a single photon of $88\text{nm}$ (really it should be $85.30\text{nm}$) has roughly enough energy to ionize nitrogen. In this case, the peak intensity of laser is less important. The only thing you need is a at least one photon each atom.
Let's put some numbers to this. So if the laser radius is $R=100\mu\text{m}$ then the cross-sectional area is roughly $\pi R^2 = 3.1\times 10^{-4}\text{cm}^2$. Let's assume your room is $L=100\text{cm}$ long (a small room). The volume of gas to ionize is $V=\pi R^2 L = 3.1\times 10^{-2}\text{cm}^3$. The atmosphere at sea level has a number density of $n_0 = 2.6\times 10^{19} \text{cm}^{-3}$. This means in the column that the laser should ionize will have $N=n_0 V=8.1\times 10^{17}\text{atoms}$. I assumed you filled your room with atomic nitrogen. Each atom requires about $I_p=14\text{eV}$ of energy so to ionize the column you will need $N I_p = 1.8\text{J}$. I switched from electron-volts to joules because eV are great for atomic energy scales and joules are great for human energy scales.
So you will need a laser with at least 1.8J of energy because no process is 100 percent efficient
Now how powerful the laser is depends on how quickly you need to ionize. For single photon ionization, you need to atoms to be ionized before the electrons recombine with the ions. I want to say recombination occurs on the nanosecond time scales at atmospheric pressures. Let's be optimistic and say it is 10ns. This means you need $E=1.8\text{J}$ in about $T=10\text{ns}$. The 10ns pulse would then and average power at least $P = E/T = 180\text{MW}$ with a radius of $100\mu\text{m}$. Is this possible at $88$nm? I do not know.
But single photon ionization is not the only way to ionize nitrogen.
There is also multiphoton ionization (MPI), where multiple photons, each with an energy too small to ionize by themselves, contribute to ionize one atom. This process depends on the intensity of the light (power per unit area). At wavelengths of $800$nm, MPI tends to be the main process in picosecond duration pulses with intensities of $I=10^{11}\text{W/cm}^2$ to $10^{12}\text{W/cm}^2$. These parameters can be reached easily with current laser systems. Notice that energy in such as pulse is $(10^{13}\text{W/cm}^2) (100\text{ps}) * \pi R^2 \approx 1\text{J} $. This is about what is needed to ionize all of the gas. To get more ionization, we can make the laser pulse longer, wavelength shorter, or intensity higher, but if we change these parameters too much other physical processes might kick in.
If the intensity does up more, then we can get tunneling ionization (TI) which is a type of photoionization. This occurs when the electric field of the laser is so strong that it pulls the electron off the atom. Really, tunneling and multiphoton ionization are different regimes of the same process but people often refer to them as different processes. In nitrogen at 800nm, this process becomes important for intensities above $10^{14}\text{W/cm}^2$ which typically occur during femtosecond laser pulses. Again these parameters can be reached with current university laser systems. The energy in one of these pulses is $(10^{15}\text{W/cm}^2) (100\text{fs}) * \pi R^2 \approx 3.1\text{J} $. Typical energies are 1mJ to 1J so my parameters may be a bit optimistic.
If you do not have enough energy to ionize all of the gas then it is very possible to partially ionize.
Finally, collisional ionization (CI) is often important for long pulses, like nanoseconds. At those durations the laser intensity can be less but you will still need at least the 1.8J to ionize most of the electrons. During collisional ionization, the few free electrons wiggle (quiver) in the electric field. If their kinetic energy is greater than the ionization energy (14eV) then when they collide with an atom they could kick off another electron.
Let me try to directly address your questions...
No. You don't need 88nm light if using MPI, TI, or CI are the processes you use. If you want to ionized each atom twice via single photon ionization, then yes you will need a shorter wavelength.
For a 1m column, 100$\mu$m wide, at least 2J per meter.
This one is tough. You need to know how much you will ionize (fully, 10%, 1%,...). This should depend on the ionization process and other laser parameters. Once you know that, then you can roughly make energy arguments about how far your pulse can go if it puts all of its energy into ionization.
If you have a particular laser you are considering than information about that laser system could help answer question 3.
Here is an image of a laser-plasma filament (link to Laser Matter Interaction group at ENSTA ParisTech. Note that laser-plasma filaments are weakly ionized and often occur around the MPI and TI regimes.
I should comment that practically creating a room-length, fully ionized, plasma column would be difficult because the plasma will have a defocusing effect on the light. Filaments avoid defocusing issue by using self-focusing to offset the plasma defocusing but this is limited to weak ionization. Plasma defocusing might make it advantageous to go to shorter wavelengths
