The formula should work if you convert the booleans (true or false) returned by LEN(A1:E)>0 to numbers (1 or 0), as Barry already mentioned. This can be done quite easily by wrapping the output of the LEN()-function in an N-function or by preceding it with '--'. So, assuming your data starts in row 2, see if this works:
=ArrayFormula(MMULT( --(LEN(A2:E)>0) , TRANSPOSE(COLUMN(A2:E2)^0)))
An alternative way would be to use COUNTIF()
=ArrayFormula(COUNTIF(IF(A2:E<>"", row(A2:A),),row(A2:A)))
and probably even a combination should work:
=ArrayFormula(MMULT( --(A2:E<>"") , TRANSPOSE(COLUMN(A2:E1)^0)))
If you also want to include a header row, try:
=ArrayFormula(if(row(A:A)=1, "Header", MMULT( --(LEN(A:E)>0) , TRANSPOSE(COLUMN(A1:E1)^0))))
or
=ArrayFormula(if(row(A:A)=1, "Header", MMULT( --(A:E<>"") , TRANSPOSE(COLUMN(A1:E1)^0))))
or
=ArrayFormula(if(row(A:A)=1, "Header", COUNTIF(IF(not(isblank(A:E)), row(A:A),),row(A:A))))
EDIT: (after new question in comments)
If you want to sum the values, you can do that with MMULT() too:
=ArrayFormula(if(row(A:A)=1, "Header", MMULT(if(A1:E<>"", A1:E,0), transpose(column(A1:E1)^0))))
or using sumif:
=ArrayFormula(if(row(A:A)=1, "Header", sumif(IF(COLUMN(A1:E1),ROW(A1:A)),ROW(A1:A),A1:E)))
NOTE: if you want to limit the output to let's say the last row that has values in col A, try:
=ArrayFormula(if(row(A:A)=1, "Header", IF(LEN(A1:A), MMULT(if(A1:E<>"", A1:E,0), transpose(column(A1:E1)^0)),)))
or, again with sumif()
=ArrayFormula(if(row(A:A)=1, "Header", if(len(A1:A), sumif(IF(COLUMN(A1:E1),ROW(A1:A)),ROW(A1:A),A1:E),)))