5 Favorite Letters

Charcoal, 51 49 bytes (ECXKI)

Ｇ←⁵↓³→⁴EＭ⁷→Ｇ↑²↗²→⁴CＭ⁶→ＧＨ↘↗³XＭ⁶→ＧＨ↓→↗³KＭ⁸→ＰTIII‖Ｏ↓

Outputs

EEEEE      CCCC     X   X     K   K     IIIII
E         C          X X      K  K        I  
EEEE      C           X       KKK         I  
E         C          X X      K  K        I  
EEEEE      CCCC     X   X     K   K     IIIII

Try it online!

Using characters that are vertically symmetrical, draws the top halves and then reflects. Most of these make use of Polygon (Ｇ) and PolygonHollow (ＧＨ) to draw a series of connected line segments with a particular character. I can more easily be done with MultiPrint (Ｐ), using T as the direction.

Note: PolygonHollow just draws the segments specified, without completing the polygon or filling it. Polygon will complete and fill (not what we want) if the polygon can be completed with a simple line in one of the eight cardinal or intercardinal directions. Otherwise, it behaves like PolygonHollow, for a savings of one byte.

The order of characters was chosen to require only horizontal moves from the endpoint of one to the start of the next. Here's how the cursor proceeds:

PowerShell v2+, 138 128 114 112 106 105 bytes (LICTD)

"L1    IIIII1 CCCC1TTTTT1DDDD
$("L11 I1  C11 T1  D   D
"*3)LLLLL1IIIII1 CCCC1  T1  DDDD"-replace1,'     '

The idea is to maximize the spaces between the letters so we can get repeated compressions.

Borrows the middle-row deduplication trick from Florent's answer. Saved 6 bytes thanks to Ben Owen by using string multiplication for the middle three rows, and an additional byte thanks to Matt.

Output is like the following at 227 bytes, for a 53.7% reduction --

PS C:\Tools\Scripts\golfing> .\5-favorite-letters.ps1
L         IIIII      CCCC     TTTTT     DDDD
L           I       C           T       D   D
L           I       C           T       D   D
L           I       C           T       D   D
LLLLL     IIIII      CCCC       T       DDDD

PHP, 107 104 102 94 86 bytes

Ok I'm confident I have the smallest possible source with this method now. I wrote a script to generate and then gzip every possible combination of five letters. There are two solutions that match for the shortest compressed -- LODIC and LDOIC. I'm going with the former because it's more fun to say.

Source:

<?=gzinflate(base64_decode('81GAA39/fwjDBQggLE8QgDCdgYDLB6EYioGqoRisHkrTSCUIEOtWAA'));

Output:

% php foo.php
L          OOO      DDDD      IIIII      CCCC
L         O   O     D   D       I       C
L         O   O     D   D       I       C
L         O   O     D   D       I       C
LLLLL      OOO      DDDD      IIIII      CCCC