Hi,
If it's only "one-of-five" and "one-of-eight" LEDs lit at the same time (as appears from the diagram), then you need only two series resistors, for example in series with the "common ground" connections to the two groups of LEDs. Another method is to mount through-hole resistors vertically, so that each consumes only around 0.02 square inches of PCB space (i.e. two "holes" on standard 0.1 inch pitch prototyping board).
A nice-looking project, and your PCB looks quite generous; on my "drawing board" I have a through-hole PCB design the size of an AA cell (i.e. 14 mm x 50 mm) which carries:
A (socketed) DIP PICaxe (08M2),
A Programming Socket (3.5 mm or the Legacy "pins"),
A Real-Time Clock module (with battery),
An I2C EEPROM (up to 512 Mb),
A 32 x 128 pixels graphic OLED display,
A LED or LDR for ambient light monitoring,
and the required "interface components and connections". All in a height only slightly larger than the diameter of an AA cell. Thus it can use a standard "AA Battery Box" (with a "window" cut for the Display), for both its "enclosure" and power supply.
Cheers, Alan.