Dysarthric speech reconstruction (DSR) aims to transform dysarthric speech into normal speech. It still suffers from low speaker similarity and poor prosody naturalness. In this paper, we propose a multi-modal DSR model by leveraging neural codec language modeling to improve the reconstruction results, especially for the speaker similarity and prosody naturalness. Our proposed model consists of: (i) a multi-modal content encoder to extract robust phoneme embeddings from dysarthric speech with auxiliary visual inputs; (ii) a speaker codec encoder to extract and normalize the speaker-aware codecs from the dysarthric speech, in order to provide original timbre and normal prosody; (iii) a codec language model based speech decoder to reconstruct the speech based on the extracted phoneme embeddings and normalized codecs. Evaluations on the commonly used UASpeech corpus show that our proposed model can achieve significant improvements in terms of speaker similarity and prosody naturalness.