The effectiveness of instruction-tuned Large Language Models (LLMs) is often limited in low-resource linguistic settings due to a lack of high-quality training data. We introduce LuxIT, a novel, monolingual instruction tuning dataset for Luxembourgish developed to mitigate this challenge. We synthesize the dataset from a corpus of native Luxembourgish texts, utilizing DeepSeek-R1-0528, chosen for its shown proficiency in Luxembourgish. Following generation, we apply a quality assurance process, employing an LLM-as-a-judge approach, retaining 227,507 high-quality instruction-answer pairs. To investigate the practical utility of the dataset, we fine-tune 14 smaller-scale LLMs ($\leq$15B parameters) on LuxIT and evaluate them on standardized Luxembourgish proficiency exams and five downstream NLP tasks. Training on LuxIT yields a mean accuracy change of +5.37 percentage points on language exams across all 14 models, with 12 of 14 showing improvement. On NLP downstream tasks, 9 of 14 models improve in macro-averaged F1, though gains on the two benchmarks do not systematically correlate. These results underscore the feasibility of leveraging monolingual synthetic data to improve LLM capabilities in low-resource languages, while highlighting the multi-faceted nature of language proficiency.