Is it permissible to continue a business partnership where half of its profit is doubtful, because the other partner paid for the machine he works with using an interest-based loan, even though the questioner's contribution was with lawful money?

Partnership is fundamentally permissible. The fact that you alone operate the machine and your partner provides the space does not affect the validity of the partnership, as long as your brother works to attract customers, and his work is in exchange for your work on the machine. It appears that the workplace is provided gratuitously; otherwise, its rent would need to be specified. As long as the company's field of work is permissible, there is no blame upon you. Your partner's taking of an usurious loan does not impact the partnership, as the prohibition is attached to his own responsibility. However, it is preferable not to partner with someone who does not avoid dealing with unlawful transactions. Partnership with a disbeliever or someone who does not refrain from usury is disliked. Dealing with someone whose wealth is a mixture of lawful and unlawful, when the exact proportions are unknown, is also disliked. This dislike strengthens or weakens according to the abundance or scarcity of the unlawful portion.