  Context. Astrometric monitoring of directly-imaged exoplanets allows the
study of their orbital parameters and system architectures. Because most
directly-imaged planets have long orbital periods (>20 AU), accurate astrometry
is challenging when based on data acquired on timescales of a few years and
usually with different instruments. The LMIRCam camera on the LBT is being used
for the LEECH survey to search for and characterize young and adolescent
exoplanets in L' band, including their system architectures. Aims. We first aim
to provide a good astrometric calibration of LMIRCam. Then, we derive new
astrometry, test the predictions of the orbital model of 8:4:2:1 mean motion
resonance proposed by Go\'zdziewski & Migaszewski, and perform new orbital
fitting of the HR 8799 bcde planets. We also present deep limits on a putative
fifth planet interior to the known planets. Methods. We use observations of HR
8799 and the Theta1 Ori C field obtained during the same run in October 2013.
Results. We first characterize the distortion of LMIRCam. We determine a
platescale and a true north orientation for the images of 10.707 +/- 0.012
mas/pix and -0.430 +/- 0.076 deg, respectively. The errors on the platescale
and true north orientation translate into astrometric accuracies at a
separation of 1 of 1.1 mas and 1.3 mas, respectively. The measurements for all
planets are usually in agreement within 3 sigma with the ephemeris predicted by
Go\'zdziewski & Migaszewski. The orbital fitting based on the new astrometric
measurements favors an architecture for the planetary system based on 8:4:2:1
mean motion resonance. The detection limits allow us to exclude a fifth planet
slightly brighter/more massive than HR 8799 b at the location of the 2:1
resonance with HR 8799 e (~9.5 AU) and about twice as bright as HR 8799 cde at
the location of the 3:1 resonance with HR 8799 e (~7.5 AU).
