Housetraining older puppy

Question
I rescued an older Boxer puppy at 4-1/2 mo. who was a pet store purchase and given up almost immediately.  She never learned to go anywhere but in her cage.  She is nearly impossible to train and I am wondering if it is going to work. I have had her for three weeks now.

I work so therefore she is in daycare with loving people every day.  She just started because crate-training is not working.  She has some digestive problems but everyone is working with this; she is underweight and small for her age.   They are teaching her to go outside at the daycare with lots of patience.  However, when she is with me I have to watch her like a hawk because she will sneak a pee and can't leave her in her crate because she will be covered with pee or poop.  She is now sleeping in bed with me and holds it all night long.  She is in my lap, on the couch, on the seat in the car, in her bed or sitting in the kitchen with me at all time. She had a terrible time a few days ago and she peed in all those places but has not since starting daycare three days ago. (She hasn't pooped in the house for that same amount of time.) I walk her and sometimes she pees as soon as we get in the house!

She is so sweet and so loving and tries very hard, I think.  She just doesn't get it.

Is there hope?

Answer
Housebreaking starts before you get home with the new puppy. If you don't have
a crate, buy one. I prefer the more enclosed, den like plastic ones. Skip the
bedding.  At first it gets wet, and later it can be chewed into choking
hazards.  A wire rack in the bottom will help keep the puppy up out of
accidents at first. They are available with the crates, but a piece of closely
spaced wire closet shelving from a home supply place is cheaper. If you
already have a metal crate, covering it may help. Just make sure you use
something the puppy can't pull in and chew. Dogs that start out in crates as
little puppies, accept them very well.  Never leave an unattended puppy loose
in the house.  If nobody can watch it, put it in the crate.  I suggest letting
the dog have its crate all its life.

Choose a command and spot you want it to use. The less accessible to strays,
the less chance of serious disease.   If it is a female, choosing a
non grassy spot will avoid brown spots later. When you bring it home, take it
to the spot and give it the command in a firm, but friendly voice. Keep
repeating the command and let the puppy sniff around. If it does anything,
praise it. Really let it know what a good dog it is and how much you love it,
and maybe a treat.  Note, being out there not only means you can praise it,
but it also keeps it from being snatched by a hawk. If it doesn't go, take it
inside and give it a drink and any meals scheduled. A young puppy will need to
go out immediately afterward. Go to the spot and follow the above routine.
Praising it if it goes is extremely important. If it doesn't go, take it back
inside and put it in its crate and try again soon. Do not let it loose in the
house until it does go.

At first it is your responsibility to know and take the puppy out when it
needs to go. It needs to go out the first thing in the morning, after eating,
drinking, and sleeping. If it quits playing, and starts running around
sniffing, it is looking for a place to go. Take it out quickly. You will just
have to be what I call puppy broke until it is a little older.

By the time most dogs are about 3 months old, they have figured out that if
they go to the door and stand, you will let them out. The praise slowly shifts
to going to the door. Some people hang a bell there for the dog to paw. If
your dog doesn't figure this out, try praising it and putting it out if it
even gets near the door. A stern "Bad dog!" is all the punishment that is
effective, and only when you catch it in the act and are sure you didn't miss
it going to the door. Clean up accidents promptly. I mostly keep the little
puppies out of the carpeted rooms. Still I need the can of carpet foam
sometimes.  First blot up all the urine you can with a dry towel.  Keep moving
it and stepping on it until a fresh area stays dry.  A couple big putty knives
work well on bowel movements.  Just slide one under it while holding it with
the other.  This gets it up with a minimum of pushing it down into the carpet.
This works with even relatively soft ones, vomit, dirt from over turned house
plants, or anything else from solids to thick liquids.  Finish up with a good
shot of carpet foam.  Note, do not let the puppy lick up the carpet foam.
Once the dog is reliably housebroken, your carpet may need a good steam cleaning.

Many people strongly strongly push cleaning up all evidence of past accidents.  I am slower to suggest that.  Dogs will return to the same spot if they can find it.  When you see one sniffing the spot, that is your clue to run it out.

Thank you for taking this puppy.  She will take some extra work at first, but you should end up with a dog you will love.