Does my amp have enough juice to power my speakers?

It is what you do with the amp.  

Keep in in mind that the #1 destroyer of speakers is distortion....and the #1 creator of distortion is trying to run an amp at or near peak output on peaky music.  Too much power is far better and cheaper than running with too little power.

Low efficiency speakers is a trap to be avoided.
Chasing your tail with low efficiency speakers limits the choices in amplifiers. The music has min 12dB peaks, speech even more. 
A low power amp will produce DC into a speaker when overdriven, really heating up the voice coils. Not good.
The differences are staggering, with your flea power amp look for min 90dB speakers, more if you can. Check out those 100dB speakers if you can. 
Just because a weak amp can dive a 85dB speaker does not mean it will do it well and be enjoyable. Adding 10dB to your speaker efficiency is likely cheaper than upgrading your amps 10dB.
Doubling your power is just 3dB. It won't cut it.
Many (most?) low power amps (tubes) use classic power supplies designed in the era of low power table top radio's and very efficient wideband speakers w/o crossovers. This concept will not work with 85dB speakers very well.
While the tubes may be able to produce power, their power supply may not. High impedace tube recitifiers, chokes and low value storage capacitors while being able to suppress 120 Hz line noise, will rapidly sag under heavy music loads. Fine for uncritical listening to scratchy 78 rpms and AM broadcast of yesteryear, but not for today's high end expectations. 
Weak power supplies will produce massive amount of its own out of phase music that will intermodulate with the signal power. Tube power supplies need to be very beefy to handle heavy and complex loads. Any voltage variation in the power supply will end up in the speakers. 
This can easily be measured. Add a resistor load to the amp and connect a coupling capacitor to the power supply and measure the signal coming out while playing various power levels. The signal can even be amplified to listening evaluation. Now add complex loads to the resistance load, like cross over capacitors and inductors and compare results.
Anything other than DC will ruin the music. The power supply should only supply DC, not music.
Think in terms of Joules in energy storage for handling peaks. Capacitance x voltage squared divided by two.
More is merrier. Low ESR is needed to kill AC noise supply power for fast peaks. 
There may be a need to spend a great deal of effort and money in the powersupply. 
Think thousands of microfarads in tube power supplies, not tens.
Inrush current limiter will be required. Been there.

udog,

I believe your Line Magnetic 216 IA can operate in triode at 22 wpc or Ultralinear at 38 wpc.  Have you tried Ultralinear operation ? Does it improve the listening experience ?  What outputs of the amp are you using?  Based on the published impedance curves it looks like the 8 ohm outputs should sound the best.  Just a couple of items that could be worthwhile trying.  

Ultimately I have found that my system can meet the demands of the speakers and can sound heavenly most of the time.  The variances are due to the quality of recordings and source inputs.  I guess I was hoping that the grass would be greener with more power.  I don't dismiss that more power might make Guns and Roses a little smoother, but for what I usually listen to, if I have a good recording, all is good.  On a separate thread I asked for advice on buying quality vinyl and I guess that is part of the equation.  Thanks for all the input.  I'm happy where I am...for now.