Step 1: The circuit and layout.
The 22uH coil consists of 35 turns of #28AWG wire over the battery, using the metal in the shell to multiply the inductance.
A simple diagram shows the placement of the parts.
Step 2: The parts list.
BC327 PNP transistor
BC337 NPN transistor
1N4148 Low-signal diode or similar (Virtually any silicon diode)
0.1uF ceramic capacitor (can be anything larger)
150pF ceramic capacitor
12uF 6v tantalum cap (can be anything over 5uF)
100K resistor (controls brightness: 470k for 50mA; 68K for 120mA)
4-ft of #28 or #30 AWG wire wrap
Perforated copper boards, pre-trimmed as shown
Step 3: Making the tube 'body'
Use the widest battery you have as the template so that the tube is loose enough for the battery to be removed and replaced.
Set aside to dry while you assemble the circuit board.