Is contributing money to a for-profit children's nursery project, for the purpose of teaching orphans for free with the profits from this contribution, considered an ongoing charity (Sadaqah Jariyah)?

There is no objection to what you are undertaking, provided that the share of profit for each partner — the merchant relative and the endowment (waqf) — is clearly defined. The endowment's share of the profit should then be disbursed for the benefit of the designated beneficiaries, such as educating needy children and similar purposes. However, if the endowment's share is defined solely as teaching children in this manner, then this involves significant ambiguity, which invalidates the partnership.