We bought a walker for my son but he used it for only 2-3 times. For one, they said it can cause "sakang" in babies and also we had a small space because we had a lot of baby stuff around the living room. What we did is we would let him walk while he's holding on to the bed frame or the couch. He would just go around it until he learned to walk on his own. They call it "gabay" and it eas very useful for me. I guess patience is really needed and you would feel panicky most of the time while doing this. Relax!
Practice makes perfect, this is key for your niece when she is at this stage. The more she is use to standing and being on her feet the more likely she will feel comfortable to take those first few steps. Give her also the most loving appreciation and caring motivations for her to develop her confidence and finally walk on her own .
Push walker or moon walkers may be of help, if you don't want to get the traditional walkers. If you really want to teach her without any baby gears at all, then just train her to walk everyday. She'll get the hang of it and eventually learn to walk on her own.