To address the resource allocation (RA) challenge in heterogeneous visible light communication (VLC) networks, this study develops an indoor VLC hybrid system comprising multiple VLC access points (APs), a single radio frequency (RF) access point, and a single infrared (IR) AP. A joint load balancing and power allocation strategy is proposed for the VLC/RF downlink, accompanied by an iterative algorithm and optimization framework tailored for the power allocation subproblem, which allocates power to each AP to maximize data throughput. The algorithm determines optimal power distribution through alternating solutions of dual variables. Furthermore, the effective capacity of a single user with varying quality of service (QoS) requirements in the IR uplink is examined, employing four distinct power control strategies: equal power allocation and Water-Filling algorithm, sub-channel independent optimization algorithm, and sub-channel joint optimization algorithm within simulations. Results indicate that this approach is agnostic to step size or initial variable values while offering enhanced convergence speed and performance compared to traditional subgradient methods. System capacity and fairness are notably improved alongside rapid convergence rates. Considering performance metrics based on system capacity, the sub-channel joint optimization algorithm is the optimal power control strategy.