The only piece we know for sure is a cryptic passage in 1903 letter from Einstein to Mileva:"I am very sorry about what has happened to Lieserl. Scarlet fever often leaves some lasting trace behind". An amateur scholar Michele Zackheim got fascinated by the incident, and did a copious amount of research over the course of five years, visiting Switzerland, Germany, England, Hungary, and Serbia, including Mileva's ancestral villages, where she talked to anyone close to her family and went through countless baptismal records and archives.
Her resulting book Einstein's Daughter: The Search for Lieserl "is a colorful glimpse of rural Serbian culture, with its patrimonial society, strong family loyalties, female subservience, slow, leisurely discourse. Zackheim does manage to eliminate a number of women as possible Lieserls, including a melodramatic Berlin actress who claimed in the 1930s to be Einstein's daughter". The rest is explicitly speculative:"a Greenwich Village painter turned writer, argues that the toddler was severely retarded and probably had Down syndrome. A simpleton child, in the language of the time, she would have been considered uneducable... Mileva, unable to place the little girl for adoption or send her to an orphanage, left her with her parents at their home in Serbia's rural Vojvodina region on the fertile Danube plain. And there Lieserl's life was poignantly short... the little girl died at 21 months after a bout of scarlet fever". Many consider this speculation plausible, but all agree that it is only a speculation.
As for the influence on Einstein's work, so little is known about the matter that it is hard to even speculate. We do know that Einstein soon entered the most productive period of his scientific life, with special relativity and photoelectric effect papers coming out in 1905. Einstein's stormy private life ("his flirtations, his stormy divorce from Mileva, his possible dalliance with the daughter of the woman who would become his second wife, his estrangement from his two sons, one of whom was schizophrenic") only came out after publication of his collected works started in 1987. One could say perhaps that he was "as roguishly independent in his personal life as he would be in his science" in early 1900s, refusing to break up with Mileva despite his mother's warning:"If she gets a child, you'll be in a pretty mess". Something more substantive emerges from the letters from the time he worked on general relativity:"His letters... show how his personal and scientific struggles intertwined in 1915, culminating in his great triumph that fall".