If the school can tell you the nature of your daughter's LD and the impact on learning, ask them to tell you what the research says about the approach they are using to teach her to read. If they do not know the answer to this question, ask them to tell you how they made the decision to use this approach. If the answer is something like: "This is the program we have for kids with LD," then ask them what the success rate is and how long they think it will take for her to learn to read better.
If they can't answer this, then ask them to track her progress on a weekly basis (looking at her acquisition and retention of very specific skills), so that you can see whether she is improving, staying the same, or getting worse. If she has not made gains after three months in the special reading program, then demand that the school increase the amount of time they work with her. Even if it means she misses some of her regular coursework. Research shows that it's the intensity of the intervention (more than the type of specialized program) that makes the difference. If your little girl does not show gains after six months in this program, then it is probably not the right program.
At this point, tell the school you want your daughter to be evaluated by a specialist outside the school. Tell them (you notice I'm not using the word ask) that if they cannot provide an intensive program of the type that the specialist recommends, you want them to find such a program, and pay for your daughter's involvement in it. Since it's the end of the school year, I would suggest finding such a program now and enrolling your daughter three to five days a week for an hour each day of intensive reading instruction, using an approach that has been shown to be effective for children with learning disabilities. This is a critically important time to do some very intense work with her. Do not wait until the fall of next year. I hope that these suggestions put your daughter on the road to success.