Automatic speech recognition (ASR) systems are well known to perform poorly on dysarthric speech. Previous works have addressed this by speaking rate modification to reduce the mismatch with typical speech. Unfortunately, these approaches rely on transcribed speech data to estimate speaking rates and phoneme durations, which might not be available for unseen speakers. Therefore, we combine unsupervised rhythm and voice conversion methods based on self-supervised speech representations to map dysarthric to typical speech. We evaluate the outputs with a large ASR model pre-trained on healthy speech without further fine-tuning and find that the proposed rhythm conversion especially improves performance for speakers of the Torgo corpus with more severe cases of dysarthria. Code and audio samples are available at https://idiap.github.io/RnV .