Bottom line:  There are no published experimental studies evaluating the effect of screening inpatients for delirium on the hospital length of stay.

Summary:
Length of stay:  A search of Joanna Briggs and of PubMed and CINAHL using combinations of these concepts–delirium, inpatients, screening, length of stay, outcomes–did not identify any studies of the impact of screening for delirium on length of stay.  There were observational studies documenting that delirium in the hospital is associated with increased length of stay.  See Han (2011), Saravay (2004) and reviews in the PubMed search.  None of these report on effect of screening on length of stay.
Delirium Screening and Assessment.  In:  Joanna Briggs JBIConnect
PubMed:  delirium[mesh] AND (inpatients OR hospitalized patients) AND screening AND length of stay

Clinical outcomes:  None of the papers identifed in the searches describe studies assessing effect of screening for delirium on other clinical outcomes, such as mortality or readmission.  Again, there are observational studies documenting the effect of delirium on these outcomes, i.e., Uthamalingam (2011), Gonzalez (2009) in the search below.
PubMed:  (“delirium/diagnosis”[mesh] OR “delirium/prevention and control”[mesh]) AND (inpatients OR hospitalized patients) AND screening AND (outcomes OR mortality OR readmission)