You don't say much about your led strip(s). Most single colour strips are 12V, and most are either 30 or 60 leds per metre. The leds are either wired in groups of 3 in series, or sometimes (especially white leds) individually. Typically each led, or group of leds, draws around 20mA to 30mA.
Your strip could be drawing as much as 10 * 60 * 30mA = 18A
or as little as 10 * (30 / 3) * 20mA = 2A
So a very wide range. Whether you would need one or two MOSFETs, and whether you would need extra heatsinks for them, depends on that total current requirement.
Also your power supply. Is it AC or DC? 12V or 24V? You said 100W. At 12V DC, that's 8.3A. So to know if the psu will run both strips, you need to know that strip current.
Can I suggest you measure the current and voltage of your strip, when lit?