Until you have multiple source notes from the mid-19th century, I'd not do it that way.
Head to www.raggedsoldier.com
and get Virginia Mescher's work on Laundry.
It's very important to remember that the original Little House books are *highly* fictionalized, and post-65; the newer "prequel" books are only slightly researched, and should be not be any kind of cue for mid-century impressions, aside from spurring a research project to see what was common, everyday, and practical for actual people in history in your region and time window.
It's also important to note that while laundry equipment is spendy for us now, it was dirt-cheap and bog-common for people then, so using atypical methods gives the wrong notions about history; even people living rough on the western trails used scrubbing boards, or just self-fabric abrasion in a creek, versus filling a tub and stomping.
While you amass more specific research, just refrain and save up for the needed equipment.