How do nicks in the DNA strand affect the success of Long Range PCR?
It is not unlikely that nicking is causing trouble. The question is if repeated freezing and thawing is causing the problem, there is an article which has been published in B.R.L. Focus in 1983 which disputes this effect. You can find the PDF with the article here, the article itself starts on page 10 in the PDF. On the other hand there is this paper ("Impact of long-term storage on stability of standard DNA for nucleic acid-based methods."), which shows that sstorage conditions do have an effect on the effciency of qPCR standards. Whatever causes nicking, it is a problem in (long) PCR. The following quote is from Sigma in their long PCR enzyme mix:


  Template High quality and adequate length of the template are
  essential for reliable amplification of larger fragments. Extreme care
  must be taken in the preparation and handling of the DNA target for
  long PCR. Nicked or damaged DNA can serve as a potential priming site
  resulting in high background. Avoid freezing, or, alternatively,
  freeze only once to minimize damage. The condition of the target DNA
  is critical. Depurination during cycling is minimized by use of
  buffers with a pH greater than 9.0 at 25 °C.


This paper ("Effect of Highly Fragmented DNA on PCR") shows that induced nicks affect the efficiency of PCR (although they do a much shorter amplification than you). So to overcome this problem I would make small aliquots of fresh DNA preps and then see, if this helps.