This Sunday was the Sikh Khalsa day parade in Toronto. A few years ago at the Sikh Khalsa celebrations, I remember a number of Anglophone politicians taking the stage and trying to outdo each other in their enthusiastic pronunciation of the phrase ਵਾਹਿਗੁਰੂ ਜੀ ਕਾ ਖਾਲਸਾ ।। ਵਾਹਿਗੁਰੂ ਜੀ ਕੀ ਫ਼ਤਹਿ ।। (vāhigurū jī kā khālasā, vāhigurū jī kī fatahi) - "the khalsa belong to God, victory belongs to God".
The English word khalsa is borrowed from Urdu, which is borrowed from Persian خالصه ḵẖāliṣah, from Arabic خالص ẖāliṣ "pure" (OED). But Hindi सिख sikh "disciple" is from Sanskrit शिष्य śiṣya from śak "to be able, to be strong" from Proto-Indo-European *ḱak- "help; be able". The Sanskrit is also the source of shakti.
The German cognate seems to be behagen "to please". Pokorny mentions some words to do with breeding, including English hatch "to produce young from an egg", but this seems phonologically doubtful.