I'm sorry that your question wasn't answered in a timely manner and I'm sorry to hear of what's going on. You need to separate the dogs and keep them separated. You need to bring your Maltese to her vet to have her wounds attended to. The prompt administration of a systemic antibiotic will prevent abscessation. You need to consult with your daughter's doctor. I expect that she too should be examined and a systemic antibiotic prescribed.
I have advanced training in canine behavior and must tell you that diagnosing and prescribing a treatment plan for intraspecies aggression should never be done in a long distance manner. If you're intent on keeping both dogs it's imperative that you have a specialist behaviorist come to your home and examine the dynamics within prior to working up a plan and then return to your home at prescribed intervals to make changes in that plan as necessary.
Dogs can be aggressive toward other dogs for many of the same reasons that they exhibit aggression toward people. Within a household, dogs generally work out their hierarchy with a minimum of injury. However, with some forms of aggression and when there is a size or strength disparity between dogs, injuries may occur and behavior modification is necessary. Dogs that have not received adequate social contact with other members of the species during early sensitive periods of development may never reliably get along with other dogs. Situations in which fights are most likely to occur are usually either competitive in nature (e.g. food, toys, resting area, access to a family member) or one in which there is high arousal (e.g. greetings - knocking at the door - territorial barking, running through the home, exiting through a door into the yard).
Female-female aggression problems are typically seen in bitches between one and three years of age. Females show the same posturing and vocal responses as males. In most cases, the problem results from an unstable social hierarchy. One dog has not deferred to the other. This is called "social status aggression". Fighting in the same household can be very difficult to safely correct. Fights are often very intense, bites may be completely uninhibited and severe injuries are common. The prognosis is guarded to fair and treatment may involve behavioral modification and psychoactive drugs. In general, female-female aggression among pets in the same household requires that you enlist the help of a veterinary behaviorist whom you can be referred to by your vet or found here: http://www.dacvb.org/ or here: apbc.org.uk or rehoming one of your dogs. The goal of training is for you to gain sufficient control over both dogs so that they no longer exhibit any inappropriate behavior toward each other. This can be achieved through reward-based training as well as desensitization and counterconditioning to stimuli and situations in which aggressive displays may be seen. In addition, it may be necessary to help support the "pack hierarchy" so that the dogs can interact appropriately even when you're not present. This would involve identifying which dog is the most likely to be dominant in the relationship and then discouraging the subordinate from challenging the dominant and encouraging to defer instead. In my opinion, this type of intraspecies aggression is far too dangerous for you to attempt to correct unaided. It's not unusual for intraspecies aggression to be redirected toward you when the level of arousal of one or both of your dogs reaches a critical threshold. Please be very careful around your dogs when they are aroused. Here is just one of many sites that discusses the manner in which a fight can be broken up safely: http://leerburg.com/dogfight.htm?set=1
Please respond with further questions or concerns if you wish.