A PICC line should, in theory, last 3 months unless it fails for some reason; i.e., blockage or vein collapse. The reason the infection is introduced through the PICC line is because it is an open wound. An IV is equally as infectious as a PICC line. An ET tube is the same. Any time a foreign body is introduced into the body it creates an area that can host germs.
My son had a PICC line for about 7 weeks and it never did need to be replaced. He remained infection free after that save for a UTI at the end of his hospital stay and the IV he needed during his 2 surgeries.
I'm sorry your daughter had to endure so may IVs! How frustrating for you. My son liked to pull his out or bend them so he ended up with a couple of them in his head.