I think this will do it.  I used three queries.

SELECT William.[Pt#], William.[Start dt], William.[End dt],
CVDate(Format(DateAdd("m",[CountNUM],[Start dt]),"yyyy/mm/""01""")) AS [Month
of stay]
FROM William, CountNumber
WHERE (((DateAdd("m",[CountNUM],[Start dt]))<=[End dt]))
ORDER BY William.[Pt#], DateAdd("m",[CountNUM],[Start dt]);

SELECT Q.[Pt#], Q.[Start dt], Q.[End dt], Q.[Month of stay], (SELECT
COUNT(*) FROM  [William-1] Q1
     WHERE Q1.[Pt#] = Q.[Pt#]
       AND Q1.[Month of stay] < Q.[Month of stay])+1 AS [Month]
FROM [William-1] AS Q
ORDER BY Q.[Pt#], Q.[Month of stay];

SELECT [William-2].[Pt#], [William-2].[Start dt], [William-2].[End dt],
[William-2].[Month of stay], [William-2].Month,
IIf([Month]=1,DateDiff("d",[Start dt],IIf([End dt]<DateAdd("m",1,[Month of
stay])-1,[End dt],DateAdd("m",1,[Month of stay])-1)),IIf([End
dt]<DateAdd("m",1,[Month of stay])-1,DateDiff("d",[Month of stay],[End
dt]),DateDiff("d",[Month of stay],DateAdd("m",1,[Month of stay])))) AS [Days
stay]
FROM [William-2];

I forgot to mention that CountNUM is a table containing numbers 0 to your
maximum.

William,

Are you counting the day they leave/arrive, it does not look like it.  There
are 31 days in January, so a patient that arrives on January 4 and doesn't
leave until March 31st would be on site for 28 days, not 27.

I've got a slightly different technique than Karl,
1.  Create a new table (I call it tbl_Numbers), with one field [Number]
which is defined as a long integer.  Fill this table with the numbers 1-12
(for the months).

2.  Create a couple of functions.  These will be used in the query to
determine the number of days the patient is in the hospital during a given
month.  These are functions I created a while ago to give me the minimum and
maximum from among a group of values passed to the function.  The nice thing
is that they work just as well for dates as it does for numbers or strings.

Public Function Minimum(ParamArray MyArray() As Variant) As Variant

   Dim intLoop As Integer

   For intLoop = LBound(MyArray) To UBound(MyArray)

       If IsEmpty(Minimum) Then
           Minimum = MyArray(intLoop)
       ElseIf IsNull(MyArray(intLoop)) Then
           'do nothing
       ElseIf MyArray(intLoop) < Minimum Then
           Minimum = MyArray(intLoop)
       End If
   Next

End Function

Public Function Maximum(ParamArray MyArray() As Variant) As Variant

   Dim intLoop As Integer

   For intLoop = LBound(MyArray) To UBound(MyArray)

       If IsEmpty(Maximum) Then
           Maximum = MyArray(intLoop)
       ElseIf IsNull(MyArray(intLoop)) Then
           'do nothing
       ElseIf MyArray(intLoop) > Maximum Then
           Maximum = MyArray(intLoop)
       End If
   Next

End Function

3.  Create the following query.  This query gives you a record for each
month that each patient is present, the patients ID number, and the number
of days they were present during the month.  You will have to play with the
parameters inside the DateDiff, Maximum, and Minimum functions to get the
values you want, which are determined by whether you are counting the start
and end dates of their stay.  Once you have the durations working out the
way you want, then modify the query and group it by the Stay_Month, Count on
the Pt_ID column, and Sum on the Duration column

SELECT tbl_Numbers.Number AS Stay_Month
            , tbl_Patient_Dates.Pt_ID
            ,
DateDiff("d",Maximum([Start_Date],DateSerial(2006,[Number],1)),
Minimum(DateSerial(2006,[Number]+1,1),[End_Date])) AS Duration
FROM tbl_Numbers, tbl_Patient_Dates
WHERE (((tbl_Numbers.Number) Between Month([Start_Date]) And
Month([End_Date]))
    AND ((tbl_Patient_Dates.Start_Date)<DateSerial(2007,1,1))
    AND ((tbl_Patient_Dates.End_Date)>DateSerial(2006,1,0)))
ORDER BY tbl_Numbers.Number;

Hope this helps.