The best solution for a multi-story house is at least one litterbox per FLOOR of house (basement counts as it's own floor too). This is for lazy cats that can't be bothered traveling the stairs to use the litterbox. Now, that is not to say that you will need to do this for her whole life, just for several months and try to wean her later (well after she heals from her surgery) and has some time to mature.
Now to figure out if she has a problem with the covered litterbox, type of litter, and/or location of litterbox (or a combination of the 3). Make sure the litter is NOT near her food/water (most cats take great offense if their food is in small proximity to their litter). Nobody likes to eat in an outhouse! Lastly make sure this is a safe place away from the dogs. This can be easily accomplished by placing a baby gate in the doorway to the "safe room" with her litterbox. In the event she is being followed by those slobering, pesky canines, she can swiftly leap across the gate to safety. More than likely her behavior will change in the next year and she won't require such high mantenance litter requirements.
Also, a note about Feliway . . . this product comes in 2 types of dispensing. The difusser plugs into the wall like a Glade plug-in air freshner to help keep calm and peace in the house (especially for anxious cats or adding a new cat to a family who already has cats). For more precise issues, such as inappropriate urination or furniture scratching . . . try the pump spray to the offended locations. This concentrated spray makes them think that they should be rubbing their cheeks on that specific area (say, your bed and couch) and not a place to put urine or scratch. Also, those Feliway diffusers are only good for 500-600 sq ft. So unless you got one for every room (who has that kind of money?) . . . the cheaper and more effective way to go would be the spray bottle.
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