The only way to confirm ovulation 100% is via ultrasound at a doctor's office. Since most women do not have access to daily ultrasounds, tracking your basal body temperature (BBT) is the next best thing. To get an accurate BBT reading, you need a BBT specific thermometer that measures to the hundredth instead of the tenth (i.e. 98.61 instead of 98.6). This is important because your temperature rises between 0.5°F and 1°F when you ovulate and maintains an elevated temperature throughout the remainder of your cycle, so you need the more specific number in order to notice the change.
It is also worth noting that a positive LH strip does not guarantee ovulation actually occurred. Your body can attempt to ovulate, produce a real LH surge, and still not successfully release an egg that cycle. Without a sustained BBT rise to confirm it, you would have no way of knowing. This is exactly why reading LH strips alongside BBT and cervical mucus gives you a much more complete picture than any single data point alone.
Once your body ovulates, progesterone begins to rise. Progesterone is the hormone responsible for the temperature increase, and it is what allows for implantation of an embryo. If no embryo implants during the luteal phase, both progesterone and estrogen decrease, triggering menstruation and the beginning of a new cycle.