The MINIMUM size cage for 1 quaker is 18 x 18 inches so that one is really on the small side...while it's tall its better to have a cage that is wider than tall. You won't find many of them though so you will want to look for something different.
The gauge of the wire is a concern as well as these guys are pretty strong and they could easily snap the welds on the wire and push them apart.
Rule of thumb is that a bird should have enough room in the cage to be able to totally spread their wings and flap without hitting anything in the cage so the bigger the better. It's said to get as large of cage as your finances will allow. A cage with bar spacing of 5/8 inch to 1 inch is recommended. While the 1 inch spread might be a little big for some quakers.
A good company for less expensive but nice cages is HQ the majority of my cages are HQ and I really like them because they have food doors that are on the sides and easily opened by humans but not so much by birds, most have easily removable bottom grates and drop pans for easy cleaning. They come in numerous styles that range from flat top to dome top to victorian or cathedral tops. They also have ones that have play tops on them.
I have one that is 26 inches wide by 20 inches deep by 30 inches tall that sets on legs so the overall height is 62 or 63 inches high bar spacing is 5/8" that would be a decent size for a pair of quakers. but again, bigger is better And it cost me about 149.00. It has a large front door for easy access etc.
Heres a couple links for the HQ cage
This is the one I have style wise
http://www.mybirdcage.com/arch_opening_cage.html but its a little smaller than mine
http://www.mybirdcage.com/ I would look in the small to medium sized cages.
While many of them have tops that open, personally I don't use that feature because it can be dangerous to the birds should the perch become loose and fall out because the hinged sides can fall and trap the bird. I just keep mine locked and hook toys to that area so the birds can't get them open.
Just some ideas for you.
Oh and one other thing you said you zip tie the doors closed. Another method that will allow you access to the doors and opening them is snap hooks. They are strong enough that quakers usually aren't able to open them but you can easily undo them to get in and out.
Good luck with your decision