To answer your last question first: yes, you can absolutely use a LIBOR-market model to get an estimate of the Value-at-Risk (VaR) of a portfolio of swaps.
To be concrete, let's suppose that you want to know the 1-year 95% VaR of a portfolio of swaps with maturity up to 5 years with pay/receive dates 3 months apart.
The steps would be:
- Pick a discretization of time. We choose equal length periods of 3 months
- Collect current curve data. Something like 3m LIBOR, 6m LIBOR, 9M LIBOR, 12 LIBOR, 2y swap, 3y swap, 5y swap
- Interpolate/bootstrap the swaps curve so that you have 3-month forward rates starting each 3 months for 5 years
- Specify the dynamics of the first four years worth of those forward rates as geometric Brownian motions (log-normal) and fit each of the relevant parameters (time-dependent drift, volatility and correlation)
- Simulate each of the forward rates for four time steps (1 year)
- Using each simulated forward curve, reprice your swaps (the longest of which now has maturity of 4 years)
- Measure VaR from your simulated swap prices
I've oversimplified a fair bit here. You might need much more precise treatment of time than my coarse "every 3 months" approach. And where the rubber really meets the road is in step 4. That's where you have to specify not only the volatility of each forward rate but the full correlation matrix between them. You might need to reach for some additional external information such as swaption prices to have something to calibrate to. If I recall correctly, the LMM does not admit a closed-form pricing formula for swaptions but pretty good approximations are available.
Also, a year might be a little too long to let the LMM run. Certainly, you wouldn't want to go much longer. The issue is that, even with a strong correlation matrix, the forward rates have nothing pulling them back towards each other and so eventually they drift off and you can get some funky forward curve shapes.
In terms of references, I don't have my copy of Brigo and Mercurio to hand but I believe it contains a fairly complete discussion of the LMM. I also found this paper useful.
The LMM can be a little intimidating at first glance. It might be worth looking first at the Black model. Once you are comfortable with it you can approach the LMM as a multi-variable version of the Black model.
Finally, to quote the preface of Brigo and Mercurio, who are themselves quoting Douglas Adams: Don't Panic.