Few things.
I was proposing using series LEDs so you can drive them without external beans, Driving them in parallel will make only 6mA for LED available and would be much trickier to set all not to blow.
LEDs can't (shouldn't, but can't) be connected in parallel, the voltage each one will drop would be slightly different for the same current, or when connecting in parallel, the current in each one would be very different, resulting in very different brightness. When you need to drive several LEDs in parallel each one should have a drop resistor to correct for this difference. Depending on how tight the specs of the LEDs being driven the smaller the resistor needs to be.
I guess you are better with 12V strips, 5V will have bigger current per section which is the limiting factor here, if you find one which could be driven fine with 20mA (one section) you may connect up to 4 sections in series with this IC.
The strips being driven for a voltage source have parallel sections of series LEDs, first datasheet popped out shows 36W for 150 LEDs at 12V, About 240mW for LED (AVG, some goes to the resistor). For the drawings seems to be you can cut them in sections of 3 LEDs, so 720mW per section, 60mA each section. If you want 2 sections you need 120mA, to have 6 LEDs, the closest to your desired 5. This happen to be bigish LEDs for a strip, 5050 size. 3528 LEDs would make 20mA per section, at 12V 20mA you could drive 3 of those with an LM391X without any trouble. If you want to drive 2 of those you are biting the edge, you could drive it to a lower current but I don't know how bright they will be, or in series but you don't have much headroom for the voltage.
At 30mA and 25V max rating you shouldn't go too close to those. Test how well the strips brights with half the rated current or 10V instead of 12V. Maybe it's enough, those things are really bright, depends on the application it might be enough. Given the added complexity you may want to go for less LEDs, dimmer LEDs, external drive, custom LED panel. I don't know if any strips goes lower than 20mA per section but in any case you will be in the same ballpark of brightness than the bigger driven softer. You could drive five 20mA LEDs in series without a resistor from the IC, the strips have internal resistors which makes it not so optimal for your case. You need to know the LED voltage and minimize the voltage in the IC for the current you want, this is probably better done empirically.
5050 datasheet:
http://www.nteinc.com/LED_tubing/pdf/LED_Strips_02.pdf3528 datasheet:
http://www.nteinc.com/LED_tubing/pdf/LED_Strips_01.pdfComprehensive datasheet:
http://www.ledlightsworld.com/datasheet/Specification-of-Flexible-LED-Strip-www.ledlightsworld.com.pdfJS