So, she respects "her space", why not mine?
Because you have only trained her to recognise those small areas of space as the areas she needs to keep clean. She doesn't see them as 'her' space but as 'your and our' space and you have successfully got her to keep herself clean in those tiny areas. But you need to start real training to extend her ability to see you want the same good behaviour in the whole house. A puppy or dog doesn't see a 'house' as a 'den' or as anywhere it needs to view differently from the grass outside or a boarding kennel run. Only you have the ability to train her to see it otherwise and this takes hard and consistent work. Most puppies naturally keep a small area clean because the mother taught them this is a whelping nest -- and this natural tendency is what you build on to extend that area of 'den' that they will keep clean. But just as the mother taught the pup to not go in the den area, so YOU have to teach her, gradually and without punishment but with constant supervision, that the house is to be kept clean as well. She will NOT make this connection on her own.
An 8 wk old puppy could simply never come to you housetrained -- this is the baby stage for dogs and the puppy will only just have begun to go on its own, outside the whelping nest! It is now just ready at this point to START the housetraining process.
Most dogs are only just beginning to become mostly reliable at 6 months and it is generally a year before they can be considered reliable. However you do get the occasional accident, still. :roll: Note that during all this time the pup is not meant to be having accidents indoors simply because it is not fully housetrained. Instead, what I am saying is that the OWNER needs to offer a very structured environment to the pup so that it never has the OPPORTUNITY to go inside.
Leaving a door open at this stage is expecting too much capability from a young pup. This is the equivalent of helping a toddler use a potty for a week or so then expecting the child top know to go use the toilet simply because the bathroom door is open. Puppies, like toddlers, have short memories and need practice, practice, practice, *in an adult's presence with adult supervision so they always 'get it right'*.
Also, they easily start to unlearn what they have learned if no one is there reinforcing the right choices.
I'd recommend ordering Shirlee Kalstone's houstraining classic Housetraining a Dog in Seven Days and using her sample schedules and very structured programme to get back on the right track. I referred to this constantly during the training period just for encouragement and advice.
Generally you need to have Lily under your eye and closely supervised at ALL times. That may mean leashing her and tying her to a belt loop. Other than that she needs to be in her crate, at arm's length with you while she plays, asleep, in her crate, or on a lap. NO EXCEPTIONS. When it is time for pottying, put her ON A LEAD and take her out. Stand with her til she goes. If she doesn't go, then supervision as above. Every single time she goes in the house is a small step back, letting her know she has a choice of where to go. She needs to never have that choice and ALWAYS be set up for success -- going outside to praise and a treat -- but the owner has the total responsibility of creating that environment for success (just like a parent or a teacher!) exhausting as it can be. Punishment should never be used -- this only makes the dog learn that going in your presence is something you don;t seem to like, so she will choose to go when you aren;t around and aren;t looking, often in hidden places.
Also note that just because a dog is housetrained to one room does not mean (and won't mean!) the pup extends that one room to mean the whole house. You need to restrict her ability to roam around, especally NEVER EVER to be out of your sight and unsupervised. You need to keep doors closed and only slowly introduce new rooms to which you go through the whole repeat process of housetraining.
It takes time and effort to housetrain -- one reason why a lot of people get adult dogs after their one introductory puppy experience! :lol: -- but the day will come when Lily can be trusted in the house.
When that moment comes will be entirely dependent on you, though, not on her.