Very High Resolution (VHR) forest structure data at individual-tree scale is essential for carbon, biodiversity, and ecosystem monitoring. Still, airborne LiDAR remains costly and infrequent despite being the reference for forest structure metrics like Canopy Height Model (CHM), Plant Area Index (PAI), and Foliage Height Diversity (FHD). We propose FSKD: a LiDAR-to-RGB-Infrared (RGBI) knowledge distillation (KD) framework in which a multi-modal teacher fuses RGBI imagery with LiDAR-derived planar metrics and vertical profiles via cross-attention, and an RGBI-only SegFormer student learns to reproduce these outputs. Trained on 384 $km^2$ of forests in Saxony, Germany (20 cm ground sampling distance (GSD)) and evaluated on eight geographically distinct test tiles, the student achieves state-of-the-art (SOTA) zero-shot CHM performance (MedAE 4.17 m, $R^2$=0.51, IoU 0.87), outperforming HRCHM/DAC baselines by 29--46% in MAE (5.81 m vs. 8.14--10.84 m) with stronger correlation coefficients (0.713 vs. 0.166--0.652). Ablations show that multi-modal fusion improves performance by 10--26% over RGBI-only training, and that asymmetric distillation with appropriate model capacity is critical. The method jointly predicts CHM, PAI, and FHD, a multi-metric capability not provided by current monocular CHM estimators, although PAI/FHD transfer remains region-dependent and benefits from local calibration. The framework also remains effective under temporal mismatch (winter LiDAR, summer RGBI), removing strict co-acquisition constraints and enabling scalable 20 cm operational monitoring for workflows such as Digital Twin Germany and national Digital Orthophoto programs.