How do we report income we made off of a sublet from an apartment (in which we have no ownership) that we rented from a landlord (with permission, of course)?

You should report your rental income in the same section where someone would if they owned their home. In both situations, the income is reported as passive income from a rental activity on schedule E.

Since you don't own the home, you won't add the house in the Asset and Depreciation section. Alternatively, you can deduct the costs of renting your home, while it was being a subleased, under the expenses section. 

Since you don't own the home, you won't be able to take a loss (even if your expenses exceed the income), but you will be able to reduce, or eliminate, the income.

To enter your rental income for your apartment that you rent:

1. Under Wages and Income, select to add income and then to see the list 
2. Scroll down to Rentals, Royalties, and Farm and click Show More
3. Select Start next to Rental Properties and Royalties (Sch E)
4. The questions in this section are directed toward a home owner, but this is the correct place to enter your rental. Answer the questions, but look out for the following three: 

• You'll be asked Was This Property Rented For All of 2017 and this is where you'll say No and enter the dates it was available to be rented and you weren't living it (which may be greater than the time you rented it)
• On the Property Ownership screen, answer No. and on the next select No, I will calculate my own allocation. This means you'll need to enter your expense totals for the time you rented the property. This is best since you don't own the property. 
• On the next screen about Active Participation, be sure to select No I am not an active participation.