Again,
It will work.
For more than just a night light, go with a 12V CFL.
The 'diffuser' at the bottom of the light tube is going to cut down the light. A LOT. Probably better to have the LEDs exposed inside the room, than inside the tube.
Go with a higher voltage, like 4 batteries instead of 2 (and if you plan on using solar garden light PVs, use 3 PV in series instead of 2, and probably want to parallel 2 strings of PVs).
With those cheap LEDs, it may be cost effective and more efficient to double the number of LEDs and run them at 10ma. More light for the same power. Or even tripple the number and run them at 6ma. Most LEDs are more efficient at 1/2ma to about 4ma.
Use wide angle (50~90') LEDs, or the light output will be either like a flash light or a disco ball.
With the garden light PVs what will make it work better is double the battery V, tripple the solar V, and (my opinion) try to use the mono or multi crystalline PVs. King size zener for a shunt charge controller.
My last toy like this uses a 12V VW PV like on ebay ($20), and 2 in series, 7.2V, 650mah, NiMh cell phone battery packs (2 for $1 surplus). Works great, but the PV is surely overkill in the amps department, and I expect it to cook the batteries if they get too low.
LEDs are great for a small amount of light, but do not expect them to light up a room. With white LEDs, in a properly designed 'bulb', somewhere around 30 LEDs and a small CFL will win on more light output for the power used.
G-