Finding current through an LED in an amplifier using a BJT
For the dc analysis, I would replace the bias by its equivalent Thévenin generator and have the following circuit:



The base current is defined as the voltage drop across the equivalent output resistance divided by \$R_{eq}\$. The voltage across the emitter resistor is the collector current plus the base current times \$R_E\$. Extract \$i_b\$ from this expression and substitute it in the first base current definition, multiply the result by the transistor gain and you have your collector current:



Looks like your dc expression has a dimension issue and Mathcad flags it.

For the ac analysis, combine \$R_A\$ and \$R_b\$ into a Thévenin generator again then scale \$V_{in}\$ accordingly and it should be easier: Thévenin and Norton are your friends in all these analyses : ) These are the first steps to the FACTs. By the way, I don't see \$r_{\pi}\$ or \$h_{11}\$ in your equivalent circuit.