Have you tried Booking.com -- http://www.booking.com/? In the yellow search box, under Guests, click on More Options. There you can put in the number of children. Another box will pop up for you to put in their ages. Note that if they are 18 or older, they count as adults.
Fill in the rest of the information to get your initial results. Then you can Filter them by using the categories down the left side of the page. Facility and Room Facility is where you will find information for your work needs. I ignore the number of stars and require that the Review Score be 8+. Then I sort them by Review Score.
I tested this for one night, 14 April, and came up with 481 lodgings in Rome City Center for 4 people, 2 adults and 2 teens ages 16 and 17. Breakfast was included for 313 of those. You need to put in your actual information and use the filters that are important to you. You will see the numbers of places diminish as you get more specific, but I'm sure that you will find many options for your family of 4.
I have never actually booked a room through Booking.com. I have only used it to identify potential lodgings, then contacted the places directly to get more detailed information on them and sometimes to get a better deal. Booking.com may say that the hotel is sold out or that there is only one room available, but that means that's all they have to sell at the moment.
You can go to Trip Advisor to see what their reviews say (knowing that these may be iffy) and to get a link to the hotel, but sometimes a link isn't provided so a Google search is in order. Most hotel websites have an English option (British or British/American flag) to click on if it comes up in the native language.
In my experiment, the results included hotel rooms, B&B's and apartments. Be aware, that requiring breakfast to be included will likely bump out the apartments. Your results are also probably going to come up with $ U.S. Dollar as the currency. In the dark blue band at the top of the screen, you can change the currency by clicking on the $ and choosing Euro or the Hotel's Currency. That's a good way to start thinking in € instead of $.
I hope this helps and that you have fun with the rest of your planning and on your trip.