Global cloud service providers handle inference workloads for Large Language Models (LLMs) that span latency-sensitive (e.g., chatbots) and insensitive (e.g., report writing) tasks, resulting in diverse and often conflicting Service Level Agreement (SLA) requirements. Managing such mixed workloads is challenging due to the complexity of the inference serving stack, which encompasses multiple models, GPU hardware, and global data centers. Existing solutions often silo such fast and slow tasks onto separate GPU resource pools with different SLAs, but this leads to significant under-utilization of expensive accelerators due to load mismatch. In this article, we characterize the LLM serving workloads at Microsoft Office 365, one of the largest users of LLMs within Microsoft Azure cloud with over 10 million requests per day, and highlight key observations across workloads in different data center regions and across time. This is one of the first such public studies of Internet-scale LLM workloads. We use these insights to propose SageServe, a comprehensive LLM serving framework that dynamically adapts to workload demands using multi-timescale control knobs. It combines short-term request routing to data centers with long-term scaling of GPU VMs and model placement with higher lead times, and co-optimizes the routing and resource allocation problem using a traffic forecast model and an Integer Linear Programming (ILP) solution. We evaluate SageServe through real runs and realistic simulations on 10 million production requests across three regions and four open-source models. We achieve up to 25% savings in GPU-hours compared to the current baseline deployment and reduce GPU-hour wastage due to inefficient auto-scaling by 80%, resulting in a potential monthly cost savings of up to $2.5 million, while maintaining tail latency and meeting SLAs.