Your dachsund most likely has gum disease if the gums are actually bleeding. She needs to be seen by a competent vet who can assess her condition. Likely she'll need to take antibiotics first to take care of the infection and then after a couple weeks (the meds need to work first) she'll need to be put under so the vet can do a thorough cleaning of her teeth. It will cost something to address your dog's issue, there's no way around that.
If you call around to smaller towns around where you live, you'll see that the further you get away from a "big" town/city, the cheaper you'll pay for services. I took both my dog and cat to a small town that's about a 45 min drive from where I live when I had to have their teeth cleaned because the price I paid to have it done was half what it cost to have the same procedure done here in Madison, WI. Small farming towns, where the vets work a lot with farmers, seem to be the best bet.
I've had the vet clean my dog's teeth a couple times. The last time I was in with the dog for her check up last fall, the vet discovered tartar build up (she actually pulled it off--yuck!) and was concerned about getting her in for another cleaning. However, I took it upon myself to brush my dog's teeth with the toothpaste I bought at the vet's office and managed to get her teeth cleaned to the point they are white and shiny again (she hates having it done. I brush every 2 to 3 nights after she eats her supper, and we play a few rounds of ball catch/chasing as a reward).
I also feed my dog very well. My miniature Schnoodle gets premium no-grain/all meat wet food (I do NOT agree with giving a dog only dry food) from a can, only premium no-grain dry dog food (not much; half of a 1/3 cup, 2x/day, mixed in with her wet food) as well as dental probiotics, vit/min supplement, Krill oil, as well as a muscle/tendon supplement (she has a trick knee) and an allergy supplement (it's helped cut her paw licking by more than half).
If you're using tap water from a town/city water tower, whether you give your dog water from the tap or from bottled water, the dog is still getting the same stuff (flouride, chlorine, other pollutants). Only by drinking well water and/or using Reverse-Osmosis water will your dog have good, clean, no-toxin water.
If you do use R/O water for your dog, though, you MUST also give your dog a good quality vitamin/mineral supplement to help replace the trace minerals that the R/O system takes out of the water. This is vitally important, as I discovered after my Schnoodle became very ill and didn't get better, despite my best intentions. It was only after I ordered some supplements for cancer protocol (yes, she was that sick but did not have cancer) and gave them to her that she got better. Then after I started her on a daily vit/min supplement it dawned on me that all of us in the household--me, my husband, my daughter, the cat (he has Feline Gingivitis/Stomatitis Complex)--we were all taking a daily vit/min, except the dog. Now we ALL take a vit/min supplement because we all drink R/O water.
Yes, it is possible to do a good tooth brushing at home, and feeding the dog good premium food and giving the dog good supplements will also help (I buy the Dental Probiotics at an upscale pet store and all the other supplements I use for the cat and dog from vitalityscience.com), but first you have to "clean up" the mess/infection. Then after her mouth's clean and free of bacteria/infection, you can start doing daily/every other day of mouth cleaning/tooth brushing. In fact, I urge every dog owner to brush the dog's teeth. You'll help prolong his/her life as well as keep costly vet dental cleaning down or nonexistent.
I hope your dachsund is healthy and happy and has perfect teeth soon.