I'm sorry to be so blunt, but anyone who can consider giving away a 10 week old puppy because it's not house trained has absolutely no idea what they're doing or how puppy training works. If you didn't want to deal with the responsibilities of an animal that pees and poops, well...
A puppy that young is just like a baby. Have you ever known of a baby who came out ready to use the toilet? no. similarly, there really is no such thing as an 8 week old puppy who comes home from the breeder's already house trained, or a 10 week old pup who's got it all down, or an older puppy who doesn't have an occasional accident.
Any puppy who is going to be living in a home needs to learn from an owner who has a complete understanding of what to do. While some pups have a better natural understanding of the system than others, it's not about the puppy figuring it out himself. It's about a consistent, positive and informed education. All of this starts with you.
I'm not sure if you got your other dog as a young puppy and trained him then, or if he came to you a bit older and already trained. In any case, I think reading at least one new, excellent puppy training book should be required for all new puppy owners. The potty training is only one small part of everything you need to be very informed about as it grows up.
On a physiological level, many puppies can't even begin to "hold it" until they're 4 months old. That's only part of what makes even considering the idea of giving her away now totally absurd. There will be a lot of ups and downs as the pup grows up, and anyone who decides to get a dog better be 100% dedicated to every responsibility that comes with it.
My absolute favorite puppy training book is
the puppy whisperer. I also have a puppy (he's 6 1/2 months old now), and before we brought him home, I went through a whole stack of puppy training books. If I could only choose one book, that would be it. Puppy and dog training in general are now based on completely different methods than used to be used. The new methods are positive, easy to learn, non-violent, extremely practical... and most importantly, they work!
My second favorite puppy training book (which might tie for first place with the one mentioned above) is called
how to raise a puppy you can live with. It's excellent. If you have the inclination, get them both.
Then, if you are looking for an extra book based completely on potty training, you can pick up a copy of
how to housebreak your dog in 7 days. It's short, cheap and extremely helpful.
If you can't get them at your local bookstore or from your local library, you can get them from
Amazon.com: Online Shopping for Electronics, Apparel, Computers, Books, DVDs & more. You can also read the book reviews there. A good book on care and training will save you more time, money and energy in the long run that you could possibly imagine.
Here are the links to these books for purchase at amazon.com (by the way, it looks like the first two are also on sale):
Amazon.com: Puppy Whisperer: A Compassionate, Non Violent Guide to Early Training and Care: Paul Owens, Terence Cranendonk, Norma Eckroate: Books
Amazon.com: How to Raise a Puppy You Can Live With: Clarice Rutherford, David H. Neil: Books
Amazon.com: How to Housebreak Your Dog in 7 Days (Revised): Shirlee Kalstone: Books
One last thing: make sure you have a special cleaning product made for removing the smell of dog urine. And use it every single time. Your pet store should have a good selection to choose from. Make absolutely sure you're not using cleaning products that contain ammonia. They smell like urine to dogs, which makes them pee in the wrong places.
Your pup will pick up all the rules. It's just a matter of you being truly informed. As her teacher, you have to tell her how things work in a way she will understand. Get the books. They will change everything given some time and patience.