I seriously doubt SpeedFan could do what you want. The problem is that the mobo's 4-pin fan port only knows how to change a fan speed by altering its PWM output signal, but the 3-pin fan you have does NOT even try to use that signal. As you have this set up, the fan will always run at full speed. Even if you plugged it into a PSU 4-pin Molex output, it will do the same - such an output only has a fixed voltage, and cannot control speed.
You have three options:
1. Buy a 4-pin fan, BUT you say that will be very difficult.
2. Check in the BIOS Setup screens for you mobo where the case fan's CHA_FAN ports are configured. SOME mobos have both 3-pin and 4-pin ports, so if you could switch the fan to a 3-pin port that could control it. SOME mobos allow you to change the way a 4-pin port operates - you can choose to use it in Voltage Control Mode (like a 3-pin port) instead of PWM Control Mode.
3. Buy a "Low-noise Adapter" for the fan. This is really just a small resistor in a plastic block that plugs into the wires between the mobo port and the fan connector. Or, it might be you get one that plugs into a PSU 4-pin Molex output and gives you a fan connector. Anyway, this unit slows the fan speed to a new constant lower setting. IF that gives you enough case cooling at the lower (and quieter) speed, you have a solution.