In the quest to improve services, companies offer customers the option to interact with agents via texting. Such contact centers face unique challenges compared to traditional call centers, as measuring customer experience proxies like abandonment and patience involves uncertainty. A key source of this uncertainty is silent abandonment, where customers leave without notifying the system, wasting agent time and leaving their status unclear. Silent abandonment also obscures whether a customer was served or left. Our goals are to measure the magnitude of silent abandonment and mitigate its effects. Classification models show that 3%-70% of customers across 17 companies abandon silently. In one study, 71.3% of abandoning customers did so silently, reducing agent efficiency by 3.2% and system capacity by 15.3%, incurring $5,457 in annual costs per agent. We develop an expectation-maximization (EM) algorithm to estimate customer patience under uncertainty and identify influencing covariates. We find that companies should use classification models to estimate abandonment scope and our EM algorithm to assess patience. We suggest strategies to operationally mitigate the impact of silent abandonment by predicting suspected silent-abandonment behavior or changing service design. Specifically, we show that while allowing customers to write while waiting in the queue creates a missing data challenge, it also significantly increases patience and reduces service time, leading to reduced abandonment and lower staffing requirements.