Thomson do a 2 bedroom apartment at the Riu Olivina Resort, Lanzarote. The Natura Palace Hotel,lanza can do a room upgrade to a family, 2 bedroom suite but it will obviously depend on your budget.

Los Zocos in Costa Teguise. I had 2 very enjoyable weeks there last summer.Rooms very spacious

In the 12 years that I have been visiting Lanzarote, I've used Ocean Club three or four times and they have a nice, neat little complex especially if you need to keep an eye on kids and yet still relax a bit. Five mins from the beach and strip if needed and yet in a quiet location.

They also have just acquired some beach side apartments which look promising if your little 'uns like the sand. I haven't stayed in these, so I cannot coment on the facilities.

I would have given you some nice easy to follow links to the above, but I seem to have breached some unwritten law about not helping people until I've got the required number of posts. If my post lives through the night, you can cut'n'paste these links into your browser, but I may have only a short time as a member of Holiday Truths............

Ocean Club Two Bed Apts - oceanclubholidays.com/site/pages/ui_properties.aspx?Complex=CO
Beach Front Apts - oceanclubholidays.com/site/pages/ui_properties.aspx?Complex=CLB

If it helps (being a long standing member lol) I can also confirm that the 2 bed apartments in the Ocean Club in Lanzarote are a good base to stay in with young kids, but I am pretty certain it doesn't do AI as requested by the OP. I have also stayed in the complex near Los Zocos think it was called Teguisol and that has 2 bed apartments and was AI. This is all years ago though when kids were younger so current reviews would need to be checked. Ocean Club was small and more intimate, and well located for all the eateries and bars in del Carmen. Teguisol was on the outskirts of the resort of Costa Teguise, walkable to a nice beach and on a larger scale altogether than Ocean Club. Both were good holidays with young kids in tow.